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VOLUME 1

GEOMETRY WITH STATISTICS

ISBN: 979-8-89353-911-0

SOUTH CAROLINA

Math Nation AGA was originally developed by Illustrative Mathematics®, and is copyright 2019 by Illustrative Mathematics. It is licensed under Creative Commons Attribution 4.0 International License (CCBY 4.0). The online curriculum resource platform oﬀered at "mathnation.com", including the videos and practice questions, as well as custom curriculum adaptations, are additions to the original Illustrative Mathematics content is copyright 2024 by Accelerate Learning, Inc.

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Table of Contents Unit 1: Foundations in Geometry .................................................. 1

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Lesson 1: Exploring Geometry Terms ..................................... 3 Lesson 2: Defined and Undefined Terms ................................13 Lesson 3: Line Segments, Rays, and Angles ...........................19 Lesson 4: Bisectors and Midpoints ........................................27 Lesson 5: Perpendicular Bisectors.........................................37 Lesson 6: Revisiting Angle Relationships ................................45 Unit 2: Rigid Transformations ..................................................... 55

Lesson 1: Isometry ............................................................. 57

Lesson 2: Rigid Transformations ........................................... 65 Lesson 3: Transformations on the Plane – Part 1 ...................... 71

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Lesson 4: Transformations on the Plane – Part 2 ...................... 79 Lesson 5: Symmetry – Part 1 .................................................. 91 Lesson 6: Symmetry – Part 2 .................................................. 95 Lesson 7: Working with Rigid Transformations ...................... 101

Lesson 8: Rigid Transformations on the Coordinate Plane ...... 109

Lesson 9: Describing Transformations on the Coordinate Plane .......................................... 115 Lesson 10: Sequences of Transformations ........................... 123 Lesson 11: Applying a Sequence of Transformations ............. 131 Lesson 12: Parallel Lines and Transversals ........................... 139 Lesson 13: Solving Problems Using Angle Pair Relationships ... 147

Table of Contents | iii

Lesson 14: Angle Relationship Proofs – Part 1 ...................... 153 Lesson 15: Angle Relationship Proofs – Part 2 ...................... 163 Unit 3: Triangle Congruence ..................................................... 173 Lesson 1: Congruent Parts ................................................. 175 Lesson 2: Congruent Triangles – Part 1.................................. 181 Lesson 3: Congruent Triangles – Part 2.................................. 189

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Lesson 4: Side-Angle-Side Congruence ................................ 197 Lesson 5: Angle-Side-Angle Triangle Congruence .................. 203 Lesson 6: Side-Side-Side Triangle Congruence...................... 209 Lesson 7: Angle-Angle-Side and Hypotenuse-Leg Congruence ............................... 215 Lesson 8: Definition of Congruence in Terms of Rigid Motions ................................................. 225

Lesson 9: Corresponding Parts of Congruent Triangles ........... 235 Lesson 10: Practicing Proofs ............................................... 245

Lesson 11: Solving Problems Using Congruence .................... 255 Unit 4: Similarity ...................................................................... 263

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Lesson 1: Dilations ........................................................... 265 Lesson 2: Dilating Lines and Angles .................................... 273 Lesson 3: Splitting Triangle Sides with Dilation – Part 1 ......... 279 Lesson 4: Connecting Similarity and Transformations ............ 285

Lesson 5: Reasoning about Similarity with Transformations .... 293 Lesson 6: Dilations on the Coordinate Plane ......................... 301 Lesson 7: Sequences that Include Dilations ......................... 311 Lesson 8: Conditions for Triangle Similarity.......................... 321 Lesson 9: Other Conditions for Triangle Similarity ................. 327 Lesson 10: Justifying Similarity .......................................... 335

iv | Table of Contents

Lesson 11: Splitting Triangle Sides with Dilation – Part 2 ....... 343 Lesson 12: Proving Triangle Similarity – Part 1 ..................... 349 Lesson 13: Proving Triangle Similarity – Part 2 ..................... 357 Lesson 14: Solving Problems Using Similarity....................... 367 Unit 5: Right Triangles and Trigonometry ................................. 377 Lesson 1: Equivalent Radical Expressions ............................ 379

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Lesson 2: Adding and Subtracting Radical Expressions .......... 387 Lesson 3: Multiplying and Dividing Radical Expressions.......... 395 Lesson 4: Proving the Pythagorean Theorem........................ 401 Lesson 5: The Converse of the Pythagorean Theorem ........... 409 Lesson 6: Finding Unknown Values in Right Triangles ............ 415 Lesson 7: Angles and Steepness ........................................ 421

Lesson 8: Half a Square .................................................... 427 Lesson 9: Half an Equilateral Triangle ................................. 435

Lesson 10: Ratios in Right Triangles.................................... 443 Lesson 11: Working with Ratios in Right Triangles................. 451

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Lesson 12: Working with Trigonometric Ratios...................... 457 Lesson 13: Applying Ratios in Right Triangles ....................... 467 Lesson 14: Sine and Cosine in the Same Right Triangle ......... 473 Lesson 15: Using Trigonometric Ratios to Find Angles ........... 479

Lesson 16: Solving Problems with Trigonometry ................... 487

Table of Contents | v

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Unit 1: Foundations in Geometry

Unit 1 | 1

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Unit 1, Lesson 1: Exploring Geometry Terms

Warm-Up: Math Talk: Transformations

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Each pair of shapes is congruent. Mentally identify a transformation or sequence of transformations that could take one shape to the other.

Guided Activity: Definitions

1. The diagram shown is a line segment.

a. Discuss with your partner why it isn’t called a line. b. Write your best definition of a line segment.

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c. Draw a line. Label the line using the letter ℎ.

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d. Write your best definition of a line.

e. Ask another partner pair the definitions they wrote. Record their definitions, and write your summary of their reasoning. You are the only person who should write in the first two columns of the table. Have a person from the other group initial next to your summary to indicate that your summary is correct. My Summary of Their Reason

Definition

My Summary of Their Reason

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A line is . . .

A line segment is . . .

Initials

Definition

A line has infinite length. Thus, any line drawn should have arrows on both ends. A line can be named by a single lowercase script letter or by any two points that lie on the line.

2. Consider the two lines shown.

a. Use the lowercase script letter 𝑚 to name the purple line on the left.

b. Use the points 𝐻 and 𝑅 to name the orange line on the right. 4 | Unit 1

A plane has infinite length and width with no height. A plane is named by a single uppercase script letter or by three points that do not lie on the same line. 3. Consider the line and points shown in a plane.

a. Complete the following steps.

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i. On a piece of tracing paper, copy 𝐾𝑇 and point 𝑃.

fold

ii. Fold the tracing paper so that the fold goes through point 𝑃 and each part of 𝐾𝑇 coincides with itself. Unfold the tracing paper, and draw a dashed line along the fold through the point, as shown.

fold

iii. Fold the dashed line on itself so that point 𝑃 coincides with 𝐾𝑇.

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iv. The resulting fold is a line that is parallel to 𝐾𝑇. Use a writing utensil and straightedge to draw the parallel line. Label the line using the letters 𝑊 and 𝐶 as shown.

v. Show that the lines are parallel by folding the tracing paper so that 𝐾𝑇 and 𝑊𝐶 coincide. Unfold the paper. Draw a dashed line along the fold, and name it 𝐵𝐻.

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fold

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b. Complete the statement. Line 𝑊𝐶 is a

rotation reflection

of 𝐾𝑇 .

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c. Work with your partner to describe how 𝐾𝑇 and 𝑊𝐶, which you drew on tracing paper, lie on the same plane.

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The table shows multiple examples of parallel lines. Description

Example of Parallel Lines C

Parallel lines 𝑇𝑊 and 𝐾𝐷 are shown on plane 𝐶.

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Parallel lines 𝑇𝑊 and 𝐾𝐷 are shown on planes 𝐵 and 𝑁.

Parallel lines 𝑇𝑊 and 𝐾𝐷 are shown on planes 𝐵, 𝐶, and 𝑁.

6 | Unit 1

Collaborative Activity: Definitions 1. Continue using the tracing paper from the previous component. a. Label a point on 𝐾𝑇 as 𝑅. Then, fold the tracing paper along 𝐵𝐻 and copy point 𝑅 to 𝑊𝐶 , naming it 𝑌. Unfold the paper, and draw a line segment that connects points 𝑅 and 𝑌.

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b. What type of angle does 𝑅𝑌 form with the dashed line? c. Use a ruler to measure the distance between point 𝑅 and the dashed line and between point 𝑌 and the dashed line. Write the measurements, with units, on the diagram.

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d. Discuss the following with your partner. When you reflect an object across a line, will corresponding points be the same distance from the line of reflection? Summarize your discussion.

2. Complete the statements.

The tracing paper is similar to a

plane. reflection.

By reflecting a line across a line of The parallel line

is is not

reflectioan, rotation,

a parallel line is created.

the same distance from the line of

reflectioan. rotation.

3. How is 𝑅𝑌 related to the dashed line?

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The table shows multiple examples of perpendicular lines. Example of Perpendicular Lines

Description

X R

Perpendicular lines 𝐾𝑇 and 𝑅𝐵 are shown on plane 𝑋.

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Perpendicular lines 𝐾𝑇 and 𝑅𝐵 are shown on planes 𝑀 and 𝑊.

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Perpendicular lines 𝐾𝑇 and 𝑅𝐵 are shown on planes 𝑀, 𝑊, and 𝑋.

4. Complete the statement.

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If two lines are perpendicular, then any two adjacent angles formed are .

Skew lines are lines that do not intersect and are not parallel.

The table shows an example of skew lines. Description

Skew lines 𝐻𝐵 and 𝐾𝑁 are shown on planes 𝐿 and 𝑆.

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Example of Skew Lines L

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Lesson Summary Points, lines, and planes are used throughout geometry. Lines are drawn with arrows on both ends to indicate that they go on forever in both directions. A line can be named by a single lowercase script letter or by any 2 points that lie on the line.

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A list of facts about lines is shown. • A line has infinite length with no height and no width. • A line is assumed to be straight. • A line is considered one-dimensional.

• Line is considered an undefined term.

• A line goes on forever in both directions.

Points, lines, and other two-dimensional figures lie on planes. A plane is named by a single uppercase script letter or by any 3 points on the plane that do not lie on the same line.

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A list of facts about planes is shown.

• A plane has infinite length and width with no height. • A plane is considered two-dimensional.

• Plane is considered an undefined term.

• A plane goes on forever.

Three types of relationships between lines were explored in this lesson: parallel lines, perpendicular lines, and skew lines. Two lines that do not intersect and lie on the same plane are called parallel lines. Two lines that do not intersect are called parallel. We can also call segments parallel if they extend into parallel lines.

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Skew lines are lines that do not intersect and are not parallel. Two lines that intersect at right angles are called perpendicular lines.

A right angle is an angle measuring exactly 90°.

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Practice Problems 1. Complete the statement.

If 2 lines intersect at right angles, the lines are

2. Lines 𝑃𝑀 and 𝑁𝑅 are shown on planes 𝑊 and 𝑌.

Which statement best describes the relationship between 𝑃𝑀 and 𝑁𝑅?

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A. Lines 𝑃𝑀 and 𝑁𝑅 are parallel lines. B. Lines 𝑃𝑀 and 𝑁𝑅 are skew lines.

C. Lines 𝑃𝑀 and 𝑁𝑅 are perpendicular lines.

D. Lines 𝑃𝑀 and 𝑁𝑅 are not parallel, perpendicular, or skew lines.

Review Problems 3. Triangle 𝑇𝐻𝐵 is a right triangle, where 𝑚∠𝐻𝑇𝐵 = 90°, 𝐻𝐵 = 37, and 𝑇𝐻 = 12. What is the length of 𝑇𝐵? 10 | Unit 1

4. Solve each equation. a. 3𝑥 − 6 = 21

b. 4𝑧 + 5 = 2 𝑧 + 23

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c. −5𝑣 + 4 + 3𝑣 = 7𝑣 − 3𝑣 − 8

Unit 1 | 11

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Unit 1, Lesson 2: Defined and Undefined Terms

Warm-Up: Circles, Circles, Circles

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What do you notice? What do you wonder?

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Exploration Activity: Definitions 1. Use two sheets of tracing paper to complete the following steps. a. Draw 𝐻𝑁 on one piece of tracing paper.

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b. Draw 𝐵𝐿 on the other piece of tracing paper.

c. Lay the piece of tracing paper with 𝐻𝑁 drawn on it on top of the piece with 𝐵𝐿 drawn on it. Place the papers so that point 𝐻 is on top of point 𝐵 and the line segments coincide. d. Place your writing utensil on endpoint 𝐻.

e. Rotate the top sheet of tracing paper until the 2 line segments make an obtuse angle.

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2. Draw the obtuse angle you created below.

3. When you rotated the top sheet, did the line segment change in size?

4. When you rotated the top sheet, did the line segment change in position?

5. Which part(s) of the obtuse angle has/have 2 points?

14 | Unit 1

6. Rotate the top sheet of tracing paper until the 2 line segments make a straight line. 7. Name the resulting line segment using the endpoints.

8. Work with your partner to estimate the number of degrees you rotated the top sheet of tracing paper from its original position on top of 𝐵𝐿.

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9. Based on the rotation and the definition of a straight angle found in the Lesson Summary, what fact about lines can be added to the “Facts about Lines” list in the previous lesson?

Guided Activity: Undefined Terms

There are terms in geometry that are considered undefined terms.

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A term is considered undefined if, when explaining the term, the explanation relies on the term itself or a close synonym. This is sometimes called a circular definition. 1. In the previous lesson, you wrote a definition for the term line. Review your definition, and determine if your definition is considered circular.

2. Discuss with your partner how the notion of distance along a line could be defined or described.

A notion such as distance along a line is considered undefined because it can be described but not defined. This notion, like other undefined terms, may be used to define other geometric terms.

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Although we have facts about lines, line is an undefined term in geometry. Other undefined terms in geometry include point and plane. Look back through the previous lesson to notice how often these terms were used without being defined. 3. Work with your partner to label each description as a line, point, or plane. Description

Undefined Term Line Point Plane

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Has two dimensions

Is often drawn as a four-sided figure

Line Point Plane

Has infinite length, no width, and no height

Line Point Plane

Has no length, no width, and no height

Line Point Plane

Named by an ordered pair in the coordinate plane

Indicates a location or position

Line Point Plane Line Point Plane

Has infinite length, infinite width, and zero height

Line Point Plane

Has no dimension or actual size

Line Point Plane

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Line Point Plane

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Lesson Summary While many terms in geometry can be defined, there are some frequently used notions that are undefined. A defined term has a definition that does not rely on using the term itself. A definition is an exact, formal statement of the meaning or description of a word. Terms that can be described but not defined are considered undefined terms.

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An example of a defined term is a straight angle. A straight angle is an angle measuring exactly 180°.

A few examples of undefined terms are line, point, and plane. Although there are facts about these terms, they can only be described; they cannot be defined.

Practice Problems

1. An image of a plane is shown.

Complete the statements to describe what is shown in the image. and

on plane

. The plane also contains

point

b. Angle

are shown

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a. Lines

and line segment

𝑀𝑋𝑌 𝑀𝑋𝑅

is a straight angle.

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. K

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2. Select all of the statements that are true.

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□ Point is an undefined term. □ A point has one dimension. □ A plane has one dimension. □ A line has no dimension or width. □ A point indicates a position or location. □ A line has no height, no width, and infinite length. Review Problems 3. Complete the statement.

If 2 lines are in the same plane and will never intersect, the lines .

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4. A sphere has a diameter of 12 inches (in.). What is the volume of the sphere, in terms of 𝜋?

5. Use the distributive property to find the product for the expression −4𝑥2(2𝑥3 − 7𝑥).

18 | Unit 1

Unit 1, Lesson 3: Line Segments, Rays, and Angles

Warm-Up: Which One Doesn’t Belong: Polygons

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Which one doesn’t belong?

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Guided Activity: Parts of Lines Examples of lines, line segments, and rays are shown in the table. Line

Line Segment

Ray

A line is an undefined term.

A part of a line that is bounded by 2 distinct end points

A part of a line that has a fixed starting point and extends infinitely in 1 direction

𝑅𝐷

𝐾𝑊

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𝐺𝐻

Line 𝑏

Ray 𝑚

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𝑇𝑀

When naming a line segment, the line segment overbar should be used. However, when talking about the length of a line segment, the overbar should not be used. 𝑇𝑀 = 12 millimeters (mm)

1. Line segment 𝑅𝐾 is shown. Use a ruler to measure 𝑅𝐾 in centimeters (cm).

𝑅𝐾 =

The ruler postulate states that the points of a line can be placed in a one-to-one correspondence with the real numbers such that the distance between 2 distinct points is the absolute value of the difference between the corresponding real numbers.

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2. Line segments 𝐾𝐶 and 𝑌𝐶 are shown.

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a. Discuss with your partner how aligning a ruler with 𝐾𝐶 illustrates the ruler postulate. Summarize your discussion.

b. Use a ruler to measure each line segment. Include units. 𝐾𝑌 =

𝑌𝐶 =

𝐾𝐶 =

c. Discuss with your partner why 𝐾𝑌 + 𝑌𝐶 = 𝐾𝐶 should be true.

3. Line segments 𝐾𝑊 and 𝑊𝑇 are shown, where 𝐾𝑊 = 4𝑥 − 7, 𝑊𝑇 = 23.5, and 𝐾𝑇 = 7𝑥 − 3.

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a. Determine the value of 𝑥.

b. What is the length of 𝐾𝑊?

Unit 1 | 21

Collaborative Activity: Segment Addition 1. Line segments 𝑌𝐵, 𝐵𝑊, and 𝑊𝐻 are shown, where 𝑌𝐵 = 2𝑥, 𝐵𝑊 = 𝑥 + 4, 𝑊𝐻 = 3𝑥 − 1, and 𝐵𝐻 = 35.

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a. Determine the value of 𝑥.

b. What is the length of 𝑌𝑊?

2. Points 𝐾, 𝐿, 𝑀, and 𝑁 are collinear such that 𝐾𝑀 = 𝐿𝑁.

a. Draw a diagram that represents points 𝐾, 𝐿, 𝑀, and 𝑁.

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b. Compare your diagram with your partner’s diagram. Discuss whether or not the diagrams drawn must all be the same. Summarize your discussion.

c. Determine the value of 𝑥 if 𝐾𝑀 = 6𝑥 − 6.2 and 𝐿𝑁 = 5𝑥 − 2.

d. Using the information you just found, determine the length of 𝐾𝑀.

22 | Unit 1

Guided Activity: Angle Addition Postulate 1. Point 𝑃 lies in the interior of ∠𝐾𝑊𝑇 as shown.

a. Draw 𝑊𝑃.

b. Use a protractor to measure each angle in degrees.

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𝑚∠𝐾𝑊𝑇 =

𝑚∠𝐾𝑊𝑃 = 𝑚∠𝑃𝑊𝑇 =

c. Discuss with your partner why 𝑚∠𝐾𝑊𝑃 + 𝑚∠𝑃𝑊𝑇 = 𝑚∠𝐾𝑊𝑇 should be true. Summarize your discussion.

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d. If 𝑚∠𝐾𝑊𝑇 = 70°, 𝑚∠𝐾𝑊𝑃 = (5𝑥 + 5)° and 𝑚∠𝑃𝑊𝑇 = (4𝑥 − 7)°, determine the value of 𝑥.

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Lesson Summary A line is a straight path that extends infinitely in both directions, represented in diagrams as a line with arrowheads at both ends. Parts of lines include line segments and rays. A ray is a set of points on a line with one endpoint. A line segment is a set of points on a line with two endpoints.

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When naming a line segment, the line segment overbar is used. However, when talking about the length of a line segment, the overbar should not be used. Examples are shown. 𝑁𝐹

Lines, line segments, and rays form angles.

𝑁𝐹 = 8.4 inches (in.)

Angles are formed wherever two lines, segments, or rays intersect.

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Angles are generally named with 3 points, where the vertex is the center letter. An angle can be named with only the letter naming the vertex, but only when it is clearly the only angle with that vertex. When naming an angle, we use the symbol ∠, which is read as “angle.” However, when talking about the measure of an angle, we use a script 𝑚 in front of the angle symbol, 𝑚∠, which is read as “the measure of angle.” Examples are shown. ∠𝑇𝐿𝑆

𝑚∠𝑇𝐿𝑆 = 72°

This lesson explored postulates that can be used to find segment lengths and angle measures.

• The ruler postulate states that the points of a line can be placed in one-to-one correspondence with real numbers such that the distance between 2 distinct points is the absolute value of the difference between the corresponding real numbers. • Points on the same line are called collinear points. The segment addition postulate states that if points 𝐴, 𝑋, and 𝐵 are collinear points, and point 𝑋 is between points 𝐴 and 𝐵, then 𝐴𝑋 + 𝑋𝐵 = 𝐴𝐵. • The angle addition postulate states that if point 𝑋 is in the interior of ∠𝐴𝑀𝐵, then 𝑚∠𝐴𝑀𝑋 + 𝑚∠𝑋𝑀𝐵 = 𝑚∠𝐴𝑀𝐵. 24 | Unit 1

Practice Problems

If 𝑚∠𝐿𝐸𝑅 = (14.35𝑥 − 9.8)°, 𝑚∠𝑊𝐸𝐿 = (2.5𝑥 + 10)°, and 𝑚∠𝑊𝐸𝑅 = 135°, what is 𝑚∠𝑊𝐸𝐿?

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2. The Sheltowee Trace Trail runs for approximately 343 miles (mi.), from Big South Fork National River in Tennessee to northern Rowan County in Kentucky. Leigh Ann created a simple map of part of the trail that goes through Natural Bridge State Resort Park.

Sand Gap Trail

22x − 0.04

Stairs on Balanced Rock Trail

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Her map is shown. Leigh Ann gave the map to hikers and told them that the total distance from Lakeside Trail Head to Sand Gap Trail is 1.05 mi.

14x + 0.12

not to scale

Complete the chart.

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1. Three angles are shown, ∠𝑊𝐸𝐿, ∠𝐿𝐸𝑅, and ∠𝑊𝐸𝑅.

Hike

Swinging Bridge 10x + 0.05 Lakeside Trail Head

Distance (Miles)

Lakeside Trail Head to Swinging Bridge

Swinging Bridge to Stairs on Balanced Rock Trail Stairs on Balanced Rock Trail to Sand Gap Trail

Unit 1 | 25

Review Problem 3. Plane 𝑃 is shown containing 𝑁𝐻 and 𝑆𝐺 .

Select all of the statements that are true.

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□ Lines 𝑁𝐻 and 𝑆𝐺 are perpendicular lines. □ Lines 𝑁𝐻 and 𝑆𝐺 are parallel lines. □ Angle 𝐺𝑌𝑆 is a straight angle. □ Angle 𝑁𝑌𝑆 is a right angle. □ Point 𝑃 is on the plane. □ Point 𝐻 is on plane 𝑃.

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26 | Unit 1

Unit 1, Lesson 4: Bisectors and Midpoints

Warm-Up: Find All the Points

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Here are 2 points labeled 𝐴 and 𝐵, and a line segment 𝐶𝐷:

1. Mark 5 points that are a distance 𝐶𝐷 away from point 𝐴. How could you describe all points that are a distance 𝐶𝐷 away from point 𝐴?

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2. Mark 5 points that are a distance 𝐶𝐷 away from point 𝐵. How could you describe all points that are a distance 𝐶𝐷 away from point 𝐵?

3. In a different color, mark all the points that are a distance 𝐶𝐷 away from both 𝐴 and 𝐵 at the same time.

Unit 1 | 27

Exploration Activity: Segment Bisector 1. Complete the following.

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a. Copy 𝑇𝐾 onto tracing paper.

b. Follow the steps on the copy of 𝑇𝐾.

i. Fold the copy of 𝑇𝐾 that is on tracing paper so the endpoints 𝑇 and 𝐾 align. ii. Place point 𝑀 on 𝑇𝐾 where the paper folds.

iii. Copy point 𝑀 onto 𝑇𝐾̅ above.

iv. Draw a line, a line segment, or a ray through point 𝑀.

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c. Discuss with your partner what you notice about 𝑇𝑀 and 𝑀𝐾.

28 | Unit 1

The midpoint of a segment divides it into 2 congruent segments. 2. Line segment 𝑊𝑋 is shown, where point 𝑃 is the midpoint.

a. Discuss with your partner what transformations can be used to show that 𝑊𝑃 maps to 𝑃𝑋.

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b. Ask 2 classmates which transformations they selected. Record their transformation and write your summary of their reason. You are the only person who should write in the first 2 columns of the table. Have the person initial next to your summary, indicating that your summary was correct.

My Summary of Their Reason

Initials

Transformation

A segment bisector is a line, ray, or line segment that cuts a line segment into 2 equal parts. 3. Line segment 𝐻𝐿 is shown, where point 𝑁 is the midpoint.

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a. Use each description to draw a segment bisector for 𝐻𝐿.

𝑵𝑲 bisects 𝑯𝑳

𝑻𝑾 bisects 𝑯𝑳

N L

b. Compare your drawings with your partner.

Unit 1 | 29

c. Using the definition of a segment bisector, discuss with your partner if all of the segment bisectors have to be at the same angle with 𝐻𝐿. Summarize your discussion.

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4. Line segment 𝑅𝐷 and 𝐻𝐺 are shown. Point 𝑇 is the point of intersection. G

a. Use tracing paper to copy 𝑅𝐷.

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b. Fold the copy of 𝑅𝐷 to determine whether point 𝑇 is a midpoint. c. Complete the statement. is is not

a segment bisector because point 𝑇

is is not

a midpoint.

Line 𝐻𝐺

30 | Unit 1

Guided Activity: Determining Lengths 1. Ray 𝐾𝐻 bisects 𝐺𝐵 at point 𝐾.

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a. Draw a diagram of the description.

b. Determine the value of 𝑥 if 𝐺𝐾 = 7𝑥 + 36 and 𝐾𝐵 = 13𝑥 − 18.

c. What is the length of 𝐺𝐵?

2. Line segments 𝑌𝐻 and 𝑀𝑉 bisect each other at point 𝑊.

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a. Draw a diagram of the description.

b. Determine the value of 𝑏 if 𝑌𝐻 = 12𝑏 − 4.5, 𝑀𝑉 = 17𝑏 + 0.75, and 𝑀𝑊 − 𝑊𝐻 = 13.25.

Unit 1 | 31

Exploration Activity: Bisecting an Angle 1. Angle 𝐹𝑋𝐺 is shown.

b. Draw 𝑋𝑌 that bisects ∠𝐹𝑋𝐺.

2. Angle 𝑅𝑇𝐻 is shown.

c. Discuss with your partner why bisecting ∠𝐹𝑋𝐺 using a protractor and a ruler may not be accurate. Summarize your discussion.

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a. Measure the angle with a protractor.

a. Use tracing paper to copy ∠𝑅𝑇𝐻.

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b. Fold the tracing paper so that 𝑇𝑅 and 𝑇𝐻 coincide. Open the paper and draw 𝑇𝑀 along the crease.

c. Explain how folding the tracing paper bisects the angle.

32 | Unit 1

Collaborative Activity: Using Angle Bisectors to Solve Problems 1. Angle 𝐻𝑀𝑋 is shown, where 𝑀𝐵 bisects ∠𝐻𝑀𝑋.

If 𝑚∠𝐻𝑀𝑋 = (7𝑥 + 3)° and 𝑚∠𝐵𝑀𝑋 = (3𝑥 + 6)°, then find the value of 𝑥.

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2. Ray 𝐷𝑌 bisects ∠𝑊𝐷𝑁 such that 𝑚∠𝑌𝐷𝑁 = (10𝑥 − 4)° and 𝑚∠𝑊𝐷𝑁 = (17𝑥 + 16.75)°.

a. Draw a diagram of the description.

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b. Yordis and Hae-Won used the given information to determine the value of 𝑥. Their equations are shown. Yordis

Hae-Won

10𝑥 − 4 = 1 (17𝑥 + 16.75)

1 (10𝑥 − 4) = 17𝑥 + 16.75 2

Discuss with your partner which student made a mistake setting up the equations. Then, write a short note to the student who made a mistake to explain their error to them.

Unit 1 | 33

3. Ray 𝐵𝑇 bisects ∠𝐻𝐵𝑁 such that 𝑚∠𝐻𝐵𝑁 = (10𝑥 − 2)°, 𝑚∠𝐻𝐵𝑇 = (5𝑥 − 1)°, and 𝑚∠𝑇𝐵𝑁 = (7𝑥 − 23)°. Find 𝑚∠𝑇𝐵𝑁.

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Lesson Summary

The midpoint of a segment divides it into 2 congruent segments. A segment bisector is a line, ray, or line segment that cuts a line segment into 2 equal parts, so a segment bisector intersects a line segment at the midpoint.

For example, 𝑍𝐻 and 𝑀𝐾 are shown. Line 𝑍𝐻 bisects 𝑀𝐾 at point 𝑅. Therefore, 𝑍𝐻 is a segment bisector, and point 𝑅 is the midpoint of 𝑀𝐾.

Since point 𝑅 is the midpoint, 𝑀𝑅 = 𝑅𝐾, and 𝑀𝐾 = 2(𝑀𝑅) = 2(𝑅𝐾).

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A line that bisects an angle is called an angle bisector. An angle bisector is a line, ray, or line segment that divides an angle into 2 equal angles.

For example, ∠𝑀𝑇𝐾 is shown, where 𝑇𝑅 bisects the angle. Therefore, 𝑇𝑅 is the angle bisector. Since 𝑇𝑅 is the angle bisector, 𝑚∠𝑀𝑇𝑅 = 𝑚∠𝐾𝑇𝑅, and 𝑚∠𝑀𝑇𝐾 = 2(𝑚∠𝑀𝑇𝑅) = 2(𝑚∠𝐾𝑇𝑅).

34 | Unit 1

Practice Problems 1. Line 𝐵𝐿 bisects 𝐻𝑌 at point 𝐶.

a. Draw a diagram of the description.

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b. Determine the value of 𝑥 if 𝐻𝐶 = 5𝑥 + 9 and 𝐶𝑌 = 8𝑥 − 36.

c. What is the length of 𝐻𝑌?

2. Line segment 𝑅𝐷 bisects 𝑇𝐶 at point 𝑋.

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a. Determine the value of 𝑥 if 𝑇𝑋 = 16 𝑥 − 41 and 𝑋𝐶 = 3𝑥 + 22. 3

b. What is the length of 𝑇𝐶?

Unit 1 | 35

3. Angle 𝐶𝑇𝑅 is bisected by 𝑇𝐻.

a. Find the value of 𝑥 if 𝑚∠𝐶𝑇𝐻 = (6𝑥 − 17)° and 𝑚∠𝐻𝑇𝑅 = (3𝑥 + 13)°.

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b. What is 𝑚∠𝐶𝑇𝑅?

Review Problems

4. Ray 𝑀𝐾 lies in the interior of ∠𝑆𝑀𝐵, as shown.

If 𝑚∠𝑆𝑀𝐾 = (15𝑥 + 7.5)°, 𝑚∠𝐾𝑀𝐵 = (−2𝑥 + 72)°, and 𝑚∠𝑆𝑀𝐵 = 125°, what is 𝑚∠𝑆𝑀𝐾?

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5. Points 𝑃, 𝐹, 𝑀, and 𝑇 are collinear such that 𝑃𝐹 = 𝑀𝑇.

a. Determine the value of 𝑥 if 𝑃𝐹 = 6𝑥 + 2 and 𝑀𝑇 = 3𝑥 + 26.3.

b. Determine the length of 𝑃𝐹.

36 | Unit 1

Unit 1, Lesson 5: Perpendicular Bisectors

Warm-Up: Two Circles

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Points 𝐴 and 𝐵 are each at the centers of circles of radius 𝐴𝐵.

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1. Compare the distance 𝐸𝐴 to the distance 𝐸𝐵. Be prepared to explain your reasoning.

2. Compare the distance 𝐹𝐴 to the distance 𝐹𝐵. Be prepared to explain your reasoning.

3. Draw line 𝐸𝐹 and write a conjecture about its relationship with segment 𝐴𝐵.

Unit 1 | 37

Exploration Activity: Exploring Perpendicular Bisectors Line segment 𝐶𝐷 is the perpendicular bisector of 𝐴𝐵, as shown.

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1. Use a protractor and ruler to find each of the measurements, in centimeters (cm). Angle ∠𝐴𝑋𝐷

∠𝐴𝑋𝐶

∠𝐵𝑋𝐷 ∠𝐵𝑋𝐶

38 | Unit 1

Measurement

Segment

Measurement

𝐴𝐵 𝐴𝑋

𝐵𝑋 𝐶𝐵

𝐷𝐵 𝐶𝐴

𝐷𝐴

2. Complete the statements. 85°. 90°. 105°.

The angle formed by the intersection of 𝐴𝐵 and 𝐶𝐷 is 2 3 4

congruent parts.

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The perpendicular bisector 𝐶𝐷 divides 𝐴𝐵 into

3. On the perpendicular bisector 𝐶𝐷, add 3 points labeled 𝑄, 𝑅, and 𝑆 in different places. 4. Use a ruler to measure the length of each line segment. Segment

𝑅𝐵 𝑆𝐴

𝑆𝐵

𝑅𝐴

Length

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5. Write a conjecture about the endpoints of a line segment and any point on its perpendicular bisector.

6. Share your conjecture with a partner. Summarize your partner’s conjecture and have them initial to indicate that your summary is correct. Partner’s Initials

Partner’s Conjecture

Unit 1 | 39

Collaborative Activity: Perpendicular Bisectors and Triangles 1. Copy 𝐴𝐵 and its perpendicular bisector 𝐶𝐷 on tracing paper.

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2. Draw 𝐴𝐷 and 𝐵𝐷.

3. Make a conjecture about ∆𝐴𝐷𝐸 and ∆𝐵𝐷𝐸.

4. Reflect ∆𝐴𝐷𝐸 over 𝐶𝐷.

5. Discuss with your partner what the reflection shows is true about ∆𝐴𝐷𝐸 and ∆𝐵𝐷𝐸. Summarize your discussion.

40 | Unit 1

6. Complete the statements. In Δ𝐴𝐵𝐷, ∠𝐷𝐴𝐵 is congruent to

∠𝐴𝐷𝐵 ∠𝐷𝐵𝐴

because a reflection is a rigid motion

that preserves angle measure. Since 𝑚∠𝐷𝐴𝐵 is equal to by definition, Δ𝐴𝐵𝐷 is a(n)

scalene isosceles equilateral

and

triangle.

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𝐴𝐷 = 𝐴𝐵, 𝐴𝐷 = 𝐵𝐷, 𝐴𝐸 = 𝐵𝐸, None of these

𝑚∠𝐴𝐷𝐵 𝑚∠𝐷𝐴𝐵

Guided Activity: Perpendicular Bisectors and Triangles 1. In the diagram, 𝑋𝑌 is the perpendicular bisector of 𝐴𝐵, 𝑋𝐴 = 7.7 millimeters (mm), 𝑌𝐵 = 6.85 mm, and 𝐴𝐵 = 7.7 mm. The figure is not drawn to scale.

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a. From the given information, label all segment measurements on the image. Then, label all other segment measurements based on the relationship of a line segment to its perpendicular bisector.

b. Given 𝑋𝑌 is the perpendicular bisector of 𝐴𝐵, mark all of the congruent angles. 2. Line segment 𝐷𝐻 is the perpendicular bisector of 𝐾𝑃, where 𝐾𝐷 = −𝑥 − 1 and 𝑃𝐷 = − 𝑥 + 19.

a. What is the value of 𝑥?

b. What is the length of 𝐾𝐷?

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Lesson Summary While exploring relationships in this lesson, you made several conjectures. A conjecture is a reasonable guess that you are trying to either prove or disprove.

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Recall that a right angle is the angle made when a straight angle is divided into 2 congruent angles. Lines that intersect at right angles are perpendicular. If such a line also divides a segment into 2 equal parts, it is called a perpendicular bisector. A perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it.

If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. A triangle formed by a line segment and 2 segments drawn from the endpoints of the segment to a point on the perpendicular bisector of the segment is an isosceles triangle.

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An isosceles triangle is a triangle that contains at least 2 equal-length sides and 2 equal interior angle measures.

A triangle formed by a line segment and another segment drawn from the endpoint of the segment to a point on the perpendicular bisector of the segment is a right triangle.

A right triangle is a triangle containing an interior right angle.

42 | Unit 1

Practice Problems 1. A sheet of paper with points 𝐴 and 𝐵 is folded so that 𝐴 and 𝐵 match up with each other.

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Explain why the crease in the sheet of paper is the perpendicular bisector of segment 𝐴𝐵. (Assume the conjecture that the set of points equidistant from 𝐴 and 𝐵 is the perpendicular bisector of segment 𝐴𝐵 is true.)

2. Line segment 𝑅𝑇 is the perpendicular bisector of 𝑈𝑆. 𝑈𝑇 = 8𝑦 − 12 and 𝑆𝑇 = 3𝑦 + 23. Find the value of 𝑦.

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3. In this diagram, line segment 𝐶𝐷 is the perpendicular bisector of line segment 𝐴𝐵. Assume the conjecture that the set of points equidistant from 𝐴 and 𝐵 is the perpendicular bisector of 𝐴𝐵 is true. Is point 𝐸 closer to point 𝐴, closer to point 𝐵, or the same distance between the points? Explain how you know. 𝐴𝐵 ⊥ 𝐶𝐷

Unit 1 | 43

Review Problems 4. Angle 𝑅𝐻𝑉 is shown, where 𝐻𝐹 bisects ∠𝑅𝐻𝑉.

b. What is the measure of ∠𝑅𝐻𝑉?

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a. Find the value of 𝑥 if 𝑚∠𝑅𝐻𝐹 = (6𝑥 + 2)° and 𝑚∠𝐹𝐻𝑉 = (4𝑥 + 10)°.

5. Marco is planning a trip from Nashville, Tennessee, to Chicago, Illinois. Marco plans on stopping twice for gas during his 472 mile (mi.) trip. He sketched the simplified map shown to represent his trip.

Stop 2

Stop 1

(4w − 13) mi.

Chicago

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(12w + 8) mi.

(15w − 19) mi.

Nashville

a. Find the value of 𝑤.

b. What is the distance between Nashville and Stop 2?

44 | Unit 1

Unit 1, Lesson 6: Revisiting Angle Relationships

Warm-Up: Which One Doesn’t Belong Four equations are shown. 𝑦=𝑥+9

𝑦 = −4𝑥 + 2

𝑦 = 3𝑥 − 2

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𝑦 = 2𝑥

1. Of the 4 equations, which one doesn’t belong?

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2. Explain your reasoning for your choice.

Unit 1 | 45

Collaborative Activity: Complementary Angles 1. Right angles 𝐷𝑅𝐾 and 𝐶𝑋𝑊 are shown. Ray 𝑅𝐿 is inside ∠𝐷𝑅𝐾, 𝑋𝐻 is inside ∠𝐶𝑋𝑊, 𝑚∠𝐷𝑅𝐿 = 28°, and 𝑚∠𝐶𝑋𝐻 = 28°. D

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a. Use the definition of complementary angles found in the Lesson Summary to explain why ∠𝐿𝑅𝐷 and ∠𝐿𝑅𝐾 are complementary angles.

b. Determine the measure of ∠𝐿𝑅𝐾.

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c. Discuss with your partner why 𝑚∠𝐻𝑋𝑊 must be the same as 𝑚∠𝐿𝑅𝐾. Summarize your discussion.

2. Angles 𝑌𝑁𝑇 and 𝐹𝐺𝐻 are shown. Ray 𝑁𝐷 is inside ∠𝑌𝑁𝑇, such that 𝑚∠𝑌𝑁𝐷 = 31°.

a. What relationship do ∠𝑌𝑁𝐷 and ∠𝐷𝑁𝑇 have?

46 | Unit 1

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b. Complete the paragraph proof. Angle 𝑌𝑁𝑇 is a(n)

acute right obtuse

angle. Therefore, by definition,

°. By the angle addition postulate, 𝑚∠𝑌𝑁𝐷 + 𝑚∠𝐷𝑁𝑇 = 𝑚∠

𝑚∠𝑌𝑁𝑇 =

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Therefore, by the substitution property of equality, 𝑚∠𝑌𝑁𝐷 + 𝑚∠𝐷𝑁𝑇 = supplementary complementary

By the definition of supplementary complementary

angles.

°.

angles, ∠𝑌𝑁𝐷 and ∠𝐷𝑁𝑇 are

c. On a piece of tracing paper, trace ∠𝑌𝑁𝐷. Determine which rigid transformations can be used to map ∠𝑌𝑁𝐷 onto ∠𝐹𝐺𝐻.

d. Use tracing paper to create ∠𝐹𝐺𝑊 so that 𝑚∠𝐹𝐺𝑊 = 𝑚∠𝑌𝑁𝑇 and 𝐺𝐻 is inside of ∠𝐹𝐺𝑊.

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e. Complete the paragraph proof.

Since 𝑚∠𝐹𝐺𝑊 = 𝑚∠𝑌𝑁𝑇 and 𝑚∠𝑌𝑁𝑇 =

, then 𝑚∠𝐹𝐺𝑊 =

by the

transitive property of equality. By the angle addition postulate, 𝑚∠𝐹𝐺𝐻 + 𝑚∠𝑊𝐺𝐻 = 𝑚∠

𝑚∠ of

+ 𝑚∠

supplementary complementary

. Therefore, by the transitive property of equality,

= 90° and are

supplementary complementary

angles by the definition

angles.

Since 𝑚∠𝑌𝑁𝐷 + 𝑚∠𝐷𝑁𝑇 = 90°, then by the transitive property of equality, 𝑚∠

+ 𝑚∠

= 𝑚∠

+ 𝑚∠

. Since 𝑚∠𝑌𝑁𝐷 = 𝑚∠𝐹𝐺𝐻, then by

the substitution property, 𝑚∠𝐹𝐺𝐻 + 𝑚∠𝑊𝐺𝐻 = 𝑚∠ 𝑚∠

= 𝑚∠

+ 𝑚∠𝐷𝑁𝑇. Therefore,

by the subtraction property of equality.

Unit 1 | 47

3. Complete the statements. Angle 𝑇𝐻𝑀 is complementary to ∠𝐿𝐺𝑋, and ∠𝑊𝐵𝐾 is complementary to ∠𝑀𝐶𝑌. Angle 𝑇𝐻𝑀 is congruent to ∠𝑊𝐵𝐾, so ∠

is congruent to ∠

4. Complete the statement.

Angle 𝑇𝐻𝑀 is complementary to ∠𝐿𝐺𝑋, and ∠𝑊𝐵𝐾 is complementary to ∠𝐿𝐺𝑋, is congruent to ∠

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so ∠

Collaborative Activity: Supplementary Angles

1. Angles 𝐺𝑊𝑁, 𝐻𝐷𝑌, and 𝐾𝐹𝑋 are shown. Angle 𝐺𝑊𝑁 is supplementary to ∠𝐻𝐷𝑌. Angle 𝐾𝐹𝑋 is supplementary to ∠𝐻𝐷𝑌. a. Determine what relationship, if any, ∠𝐺𝑊𝑁 and ∠𝐾𝐹𝑋 have.

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b. Discuss with your partner what can be concluded about ∠𝐺𝑊𝑁 and ∠𝐾𝐹𝑋. Summarize your discussion.

48 | Unit 1

c. Ask 2 other classmates what they concluded about ∠𝐺𝑊𝑁 and ∠𝐾𝐹𝑋. Record their conclusions, and write your summary of their reasons. You are the only person who should write in the first two columns of the table. Have someone from the other group initial next to your summary to indicate that your summary is correct. My Summary of Their Reason

Initials

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Conclusion

2. Complete the statements.

d. Discuss your classmates’ conclusions with your partner.

Angle 𝐻𝑇𝑀 is supplementary to ∠𝐿𝑋𝐺, and ∠𝐵𝐾𝑊 is supplementary to ∠𝑀𝐶𝑌.

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Angle 𝐿𝑋𝐺 is congruent to ∠𝐵𝐾𝑊, so ∠

3. Complete the statement.

is congruent to ∠

Angle 𝐻𝑇𝑀 is supplementary to ∠𝐿𝑋𝐺, and ∠𝐵𝐾𝑊 is supplementary to ∠𝐿𝑋𝐺, is congruent to ∠

so ∠

Unit 1 | 49

Guided Activity: Proving Angle Relationships Two adjacent angles that form a straight angle are called a linear pair. 1. Use the phrases in the word bank to complete the two-column proof. Some phrases may be used more than once.

Definition of linear pair Supplements of the same angle are congruent. Given: ∠2 and ∠3 form a linear pair. ∠3 and ∠4 form a linear pair.

Linear pair postulate

Complements of the same angle are congruent.

Prove: ∠2 ≅ ∠4

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Word Bank

Statement

1. Given

3. ∠2 and ∠3 are supplementary angles.

5. ∠2 ≅ ∠4

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1. ∠2 and ∠3 form a linear pair.

2. Given

4. ∠3 and ∠4 are supplementary angles.

2. ∠3 and ∠4 form a linear pair.

Reason

2. Angle 2 and ∠4 are also known as

50 | Unit 1

angles.

Lesson Summary This lesson revisited several angle relationships first explored in a prior grade. Two angles that can be combined to form a right angle are called complementary angles.

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Complementary angles are two angles with measures that sum to 90°.

Examples of complementary angles are shown. • The congruent complements theorem states that complements of congruent angles are congruent.

30°

38°

45°

60°

45°

52°

A corollary is a true statement that is a simple deduction from a theorem or postulate. Its proof requires only a few simple statements in addition to the proof of the original theorem or postulate.

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• The corollary to congruent complements theorem states that complements of the same angle are congruent. Two angles that can be combined to form a straight angle are called supplementary angles.

Supplementary angles are two angles with measures that sum to 180°.

Examples of supplementary angles are shown. • The congruent supplements theorem states that supplements of congruent angles are congruent. • The corollary to congruent supplements theorem states that supplements of the same angle are congruent. © Accelerate Learning Inc. - All Rights Reserved

55° 125°

152° 28°

Unit 1 | 51

A linear pair is 2 adjacent angles formed by 2 intersecting lines. By the linear pair postulate, if 2 angles form a linear pair, then they are supplementary.

Adjacent angles share a side and a vertex.

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Intersecting lines form pairs of vertical angles.

Vertical angles are opposite angles formed when two lines intersect.

Vertical angles are across the intersection point from each other, as shown. Pairs of vertical angles created by the intersection of 2 lines are shown in different colors.

147°

33°

147°

By the vertical angles theorem, if 2 lines intersect, then the vertical angles are congruent.

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Practice Problems

1. Right angles 𝑋𝑃𝐹 and 𝐵𝑊𝑌 are shown. Ray 𝑃𝐻 is inside ∠𝑋𝑃𝐹, such that 𝑚∠𝐻𝑃𝐹 = 39° and 𝑚∠𝐻𝑃𝐹 = 𝑚∠𝐾𝑊𝑌. B What is 𝑚∠𝐵𝑊𝐾?

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H X F P

52 | Unit 1

2. Three angles are shown with the given information. Angle 𝐴 is supplementary to ∠𝐷.

Angle 𝐿 is supplementary to ∠𝐷.

(2x + 18)°

(3z + 11)°

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What is the value of 𝑧?

(x − 33)°

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Review Problems

3. Lines 𝐻𝑋 and 𝑀𝐵 are shown. Point 𝑊 is the intersection point of 𝐻𝑋 and 𝑀𝐵 . Find the measure of ∠𝐻𝑊𝑀 if 𝑚∠𝐻𝑊𝐵 = (25𝑥 + 2)° and 𝑚∠𝐵𝑊𝑋 = (20𝑥 − 2)°.

4. Line segment 𝑃𝐾 is the perpendicular bisector of 𝑅𝑁, where 𝑅𝑃 = 9𝑥 − 5.1 and 𝑃𝑁 = 6𝑥 + 1.2, as shown. Find the value of 𝑥.

Unit 1 | 53

5. Angle 𝐻𝑇𝑁 is bisected by 𝑇𝑅.

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b. What is the measure of ∠𝐻𝑇𝑁?

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a. Find the value of 𝑥 if 𝑚∠𝐻𝑇𝑅 = (7𝑥 − 8.5)° and 𝑚∠𝑅𝑇𝑁 = (5𝑥 + 3.5)°.

54 | Unit 1

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Unit 2: Rigid Transformations

Unit 2 | 55

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Unit 2, Lesson 1: Isometry

Warm-Up: Notice and Wonder: Two Triangles and an Arrow

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What do you notice? What do you wonder?

Unit 2 | 57

Exploration Activity: Isometry Work with your partner to complete the following. 1. Five frames are shown that feature a polygon in different positions.

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Describe how the polygon moves from one frame to the next. Description

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Movement from . . . frame 1 to frame 2

frame 2 to frame 3

frame 3 to frame 4

frame 4 to frame 5

58 | Unit 2

2. The diagram shows the position of the polygon from frame 1.

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a. Label 2 pieces of tracing paper as A and B. Then, complete the steps for each piece of tracing paper. Tracing paper A • Copy the polygon.

• Move tracing paper A to display the polygon as it is shown in frame 2. Tracing paper B

• Copy the polygon and the square that surrounds it.

• Fold the tracing paper vertically so that the top corners of the square match up.

• With the tracing paper folded, draw the copy of the polygon on the other half of the tracing paper.

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b. Complete the statements.

To draw the polygon as it is shown in frame 2, tracing paper A was

moved

up. left. right. down.

When tracing paper B was folded, the polygon could be traced so that the new polygon is as shown in frame 2. The original polygon was

flipped across rotated around

The fold of the paper was the

the fold of the paper.

center of rotation. line of reflection. Unit 2 | 59

3. The diagram shows where the polygon is in frame 2. a. Complete the steps to find the point of rotation. • Label a piece of tracing paper as C. • Copy the polygon onto tracing paper C.

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• Use the point of your writing utensil to find the point of rotation so the polygon can be rotated to look like frame 3. • Place a dot on frame 2 to show the point of rotation.

• Place tracing paper C on the circular graph paper so the point of rotation is where the two red axes meet, and the polygon is oriented as in frame 2. • Rotate tracing paper C so that it looks like frame 3.

60°

30°

150°

90°

120°

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180°

210°

240°

330° 270°

300°

b. Complete the statements.

To move the polygon so it looks like frame 3, rotate the diagram on tracing paper C

degrees clockwise

The center of rotation is

60 | Unit 2

degrees counterclockwise

4. Use tracing paper, a ruler, or a protractor as necessary to complete the following steps. a. Complete the table by determining if the polygon changed in size or shape as it moved. Movement from . . .

Did the size change?

Did the shape change?

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frame 1 to frame 2

frame 2 to frame 3

frame 3 to frame 4

frame 4 to frame 5

b. What tools or information were used to determine if the size changed?

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c. What tools or information were used to determine if the shape changed?

d. Discuss with your partner whether the measurement of angles in the polygon changed from frame 1 to frame 5.

Unit 2 | 61

Collaborative Activity: Isometry 1. A diagram of 𝑅𝑇𝐷𝐻 and 𝑅′𝑇′𝐷′𝐻′ is shown. Arrows are drawn from vertex 𝐷 to vertex 𝐷′.

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𝑅′𝑇′𝐷′𝐻′ is read as “𝑅 prime, 𝑇 prime, 𝐷 prime, 𝐻 prime.” This notation is often used when a figure has been transformed.

D''

The arrows show a possible path of movement. For the movement to be correct, the same path should exist for all points on a diagram.

Discuss with your partner whether all of the points on 𝑅𝑇𝐷𝐻 can be moved using the same path as point 𝐷.

These arrows represent a mapping. A mapping is a standardized form of showing a transformation.

2. A diagram of 𝑅𝑇𝐷𝐻 and 𝑅′𝑇′𝐷′𝐻′ is shown. Arrows are drawn from vertex 𝐷 to point 𝐾.

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a. Draw a line segment from point 𝐾 to line 𝑛. What is the length of the line segment?

b. Draw a line segment from point 𝐷′ to line 𝑛. What is the length of the line segment?

D''

H D

c. Describe the relationship that each line segment has to line 𝑛.

62 | Unit 2

If corresponding points of reflected figures are the same distance from a line, then the line is the line of reflection. d. For each of the other vertices on 𝑅𝑇𝐷𝐻, draw a new point using the same movements that were used to map point 𝐷 to point 𝐾. Then, show the reflection to 𝑅′𝑇′𝐷′𝐻′ using line segments.

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e. Discuss with your partner whether all of the points on 𝑅𝑇𝐷𝐻 can be mapped using the same path and reflection or whether the mapping only works for the vertices. Summarize your discussion.

Lesson Summary

The movements explored in this lesson are known as reflections, rotations, and translations. At some points during the lesson, a reflection was referred to as a flip, and a translated figure was referred to as being moved, which are informal terms.

A reflection is a transformation that produces the mirror image of a geometric figure across a line of reflection.

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A rotation is a transformation in which a figure is turned about a center point or axis. The amount of rotation can be expressed in the number of degrees. For two- dimensional figures, the direction of the rotation can be expressed as clockwise or counterclockwise. A translation is a transformation in which every point in a figure is moved in the same direction and by the same distance.

An isometry is a transformation that preserves distance and angle measure. Reflections, rotations, and translations are examples of isometry because all 3 transformations preserve distance and angle measures. They are also referred to as rigid transformations. A rigid transformation is a transformation of points in space consisting of a sequence of one or more translations, reflections, or rotations. Rigid transformations preserve distances and angle measures.

Unit 2 | 63

Practice Problems

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1. Here are 4 triangles that have each been transformed by a different transformation. Which transformation is not a rigid transformation?

Review Problems

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2. Points 𝐸, 𝐵, and 𝐶 are collinear. Explain why points 𝐴, 𝐵, and 𝐷 are collinear.

3. What is the measure of ∠𝐴𝐵𝐸?

64 | Unit 2

Unit 2, Lesson 2: Rigid Transformations

Warm-Up: Notice and Wonder: Transformed What do you notice?

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What do you wonder?

Unit 2 | 65

Exploration Activity: What’s the Same?

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This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

Draw each rigid transformation. Use the Style Bar to choose a different color for each one.

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1. Translate figure 𝑆 along the line segment 𝑣 in the direction shown by the arrow. Color:

2. Reflect figure 𝑆 across line 𝑦. Color:

3. Reflect figure 𝑆 across line 𝑚. Color:

4. Translate figure 𝑆 along the line segment 𝑤 in the direction shown by the arrow. Reflect this image across line 𝑦. Color:

5. How are the images the same? How are they different?

66 | Unit 2

Collaborative Activity: Does Order Matter? This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

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Here is an applet with 3 congruent L shapes on a grid.

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1. Describe a sequence of transformations that will take Figure 𝐴 onto Figure 𝐵. 2. If you reverse the order of your sequence, will the reverse sequence still take 𝐴 onto 𝐵?

3. Describe a sequence of transformations that will take Figure 𝐴 onto Figure 𝐶.

4. If you reverse the order of your sequence, will the reverse sequence still take 𝐴 onto 𝐶?

Unit 2 | 67

Lesson Summary A figure is said to be congruent to another figure if there is a sequence of translations, rotations, and/or reflections that takes 1 of the figures onto the other.

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Figures that have exactly the same shape and size are congruent. Equivalently, two figures are congruent if one can be mapped to the other using a rigid transformation. This is because translations, rotations, and reflections are rigid motions. Any sequence of rigid motions is called a rigid transformation. A rigid transformation is a transformation that doesn’t change the measurements of any figure. With a rigid transformation, figures like polygons have corresponding sides of the same length and corresponding angles of the same measure. The result of any transformation is called the image.

An image is the result of translations, rotations, and reflections on an object. Every part of the original object moves in the same way to match up with a part of the image.

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The points in the original figure, called the preimage, are the inputs for the transformation sequence and are named with capital letters. The points in the image are the outputs and are named with capital letters and an apostrophe, which is referred to as “prime.” There are many ways to show that 2 figures are congruent, since many sequences of transformations take a figure to the same image. However, order matters in a set of instructions. Sometimes 2 steps in a sequence can be switched and can result in the same output, but other times, switching 2 steps results in a different image.

The 2 sequences of transformations shown both have the points 𝐴, 𝐵, and 𝐶 as inputs and points 𝐴′, 𝐵′, and 𝐶′ as outputs. Each step in the sequences of rigid transformations creates a triangle that is congruent to ∆𝐴𝐵𝐶.

68 | Unit 2

Practice Problems

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1. Here are 4 triangles that have each been transformed by a different transformation. Which transformation is not a rigid transformation?

2. What is the definition of congruence?

A. If two figures have the same shape, then they are congruent.

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B. If two figures have the same area, then they are congruent. C. If there is a sequence of transformations taking one figure to another, then they are congruent. D. If there is a sequence of rotations, reflections, and translations that take one figure to the other, then they are congruent.

3. There is a sequence of rigid transformations that takes 𝐴 to 𝐴′, 𝐵 to 𝐵′, and 𝐶 to 𝐶′. The same sequence takes 𝐷 to 𝐷′. Draw and label 𝐷′.

Unit 2 | 69

Review Problems 4. In this diagram, line segment 𝐶𝐷 is the perpendicular bisector of line segment 𝐴𝐵.

Assume the conjecture that the set of points equidistant from 𝐴 and 𝐵 is the perpendicular bisector of 𝐴𝐵 is true. Is point 𝑀 closer to point 𝐴, closer to point 𝐵, or the same distance from both points? Explain how you know.

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𝐴𝐵 ⊥ 𝐶𝐷

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a. (4𝑚3)(2𝑚2𝑛)5

5. For each expression, apply exponent laws to generate an equivalent expression with the fewest factors possible.

21𝑥4𝑦7

b. 6𝑥9𝑦2

𝑤4𝑦𝑧−2 3 � 𝑤−1𝑦𝑧5

c. �

70 | Unit 2

Unit 2, Lesson 3: Transformations on the Plane – Part 1

Warm-Up: Left to Right

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The semaphore alphabet is a way to use flags to signal messages. Here’s how to signal the letters Z and J. For each, precisely describe a rotation that would take the left-hand flag to the right-hand flag. J

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Unit 2 | 71

Exploration Activity: Turning on a Grid

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This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

2. Rotate 𝐴𝐵𝐶𝐷 180° around 𝑅.

1. Rotate 𝐴𝐵𝐶𝐷 90° degrees clockwise around 𝑄. 3. Rotate 𝐻𝐽𝐾𝐿𝑀𝑁 120° clockwise around 𝑂.

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4. Rotate 𝐻𝐽𝐾𝐿𝑀𝑁 60° counterclockwise around 𝑃.

72 | Unit 2

Guided Activity: Transformations on the Plane 1. The diagram shows ∆𝑅𝐾𝑋 and ∆𝑅′𝐾′𝑋′ on circular grid paper. Follow the steps shown.

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• Use a piece of tracing paper to copy ∆𝑅𝐾𝑋 and point 𝑀.

• Place the tracing paper so that it aligns with ∆𝑅𝐾𝑋 and point 𝑀. • Place the tip of your writing utensil on point 𝑀, and rotate ∆𝑅𝐾𝑋 so that it lies on top of ∆𝑅′𝐾′𝑋′.

b. Draw ∠𝐾𝑀𝐾′.

a. Draw ∠𝑅𝑀𝑅′.

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c. Complete the table.

Angle

Measurement (degrees)

∠𝑋𝑀𝑋′ ∠𝑅𝑀𝑅′

∠𝐾𝑀𝐾′

d. Discuss with your partner how the angle measures on the circular grid can be used to verify the rotation and how to determine the direction of rotation. Summarize your discussion.

Unit 2 | 73

e. Identify the transformation of ∆𝑅𝐾𝑋. Describe the transformation using direction or degrees, if needed. Type of Transformation

Reflection Translation

Degrees

Center of rotation is

Line of reflection is

Down Left Right Up

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Rotation

Center, Line, or Movement

f. Discuss with your partner whether the transformation could be described as “a rotation of 270° clockwise, centered at point 𝑀.” Explain why or why not.

2. The diagram shows ∆𝑅𝐾𝑋 and ∆𝑅′𝐾′𝑋′ without the circular grid paper.

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a. Draw ∠𝑋𝑀𝑋′, ∠𝑅𝑀𝑅′, and ∠𝐾𝑀𝐾′.

b. Discuss with your partner how the degree of rotation can be determined using the angles that connect the vertices and the center of rotation.

74 | Unit 2

R' R' X' X'

K' K'

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3. The diagram shows ∆𝑅𝐾𝑋 and ∆𝑅′𝐾′𝑋′ using a different transformation.

a. Identify the transformation of ∆𝑅𝐾𝑋. Describe the transformation using direction or degrees, if needed.

Reflection

Center of rotation is

Line of reflection is

Down Left

Right

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Translation

Degrees

Rotation

Center, Line, or Movement

Type of Transformation

b. Two students’ descriptions of the transformation of ∆𝑅𝐾𝑋 are shown.

• Use a piece of tracing paper to copy ∆𝑅𝐾𝑋.

Michaela

Marc

A reflection across a line through point 𝑀

A translation down

• Perform each student’s transformation to prove that neither transformation works.

c. Discuss with your partner why both students are wrong. Write your reasoning for why the transformations do not work.

Unit 2 | 75

d. The diagram shows 𝑇𝐾𝐹𝑌 and 𝑇′𝐾′𝐹′𝑌′.

Identify the transformation of 𝑇𝐾𝐹𝑌. Describe the transformation using direction or degrees, if needed.

Y K' T' F

Type of Transformation

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Center, Line, or Movement

Rotation

Center of rotation is

Line of reflection is

Down

Reflection

Left

Translation

Right

Lesson Summary

Degrees

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A rotation can be described using a center, an angle, and a direction (clockwise or counterclockwise). To perform a rotation, all 3 pieces of information need to be known. Two examples of rotations of a figure are shown and described.

Rotate 𝐴𝐵𝐶𝐷 90° clockwise about point 𝑃.

76 | Unit 2

Rotate 𝐸𝐹𝐺 120° counterclockwise about point 𝐶.

Practice Problems 1. The table shows 2 different rotations of 𝑇𝐾𝐹𝑌. Determine the direction of rotation, center of rotation, and the degree of rotation for each. Transformation B

• Direction of rotation is

. .

• Degree of rotation is

• Direction of rotation is • Center of rotation is

• Degree of rotation is

• Center of rotation is

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Transformation A

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2. Which segment is the image of 𝐴𝐵 when rotated 90° counterclockwise around point 𝑃?

3. There is an equilateral triangle, 𝐴𝐵𝐶, inscribed in a circle with center 𝐷. What is the smallest angle you can rotate triangle 𝐴𝐵𝐶 around 𝐷 so that the image of 𝐴 is 𝐵? A. 60° B. 90°

Unit 2 | 77

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4. Draw the image of quadrilateral 𝐴𝐵𝐶𝐷 when rotated 120° counterclockwise around the point 𝐷.

Review Problems

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□ A □ B □ C □ D □ E

5. Select all the points that stay in the same location after being reflected across line ℓ.

6. Two distinct lines, ℓ and 𝑚, are each perpendicular to the same line 𝑛.

a. What is the measure of the angle where line ℓ meets line 𝑛?

b. What is the measure of the angle where 𝑚 line meets line 𝑛?

78 | Unit 2

Unit 2, Lesson 4: Transformations on the Plane – Part 2

Warm-Up: Which One Doesn’t Belong: Crossing the Line Which one doesn’t belong?

Figure 4

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Figure 3

Figure 2

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Figure 1

Unit 2 | 79

Exploration Activity: Transforming Triangles

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This activity requires the use of an applet, so please make your way over to the digital platform to find both links.

1. Translate ∆𝐴𝐵𝐶 by the directed line segment from 𝐴 to 𝐶.

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a. What is the relationship between 𝐵𝐶 and 𝐵′𝐶′? Explain your reasoning.

b. How does the length of 𝐵𝐶 compare to the length of 𝐵′𝐶′? Explain your reasoning.

80 | Unit 2

2. Translate 𝐷𝐸 by directed line segment 𝑤. Label the new endpoints 𝐷′ and 𝐸′. a. Connect 𝐷 to 𝐷′ and 𝐸 to 𝐸′.

3. Marleigh started reflecting ∆𝐶𝐷𝐸 across line 𝑚. So far, she knows that the image of 𝐷 is 𝐷′ and the image of 𝐸 is 𝐸′.

a. Reflect 𝐶 to create the image of 𝐶′.

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b. What kind of shape did you draw? What properties does it have? Explain your reasoning.

b. Connect the points 𝐶′, 𝐷′, and 𝐸′ to make ∆𝐶′𝐷′𝐸′.

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c. How does the distance between each vertex on ∆𝐶𝐷𝐸 and line 𝑚 compare to the distance between each vertex on ∆𝐶′𝐷′𝐸′ and line 𝑚?

Unit 2 | 81

Guided Activity: Determining a Single Transformation 1. The diagram shows 𝑉𝑊𝑋𝑌 and 𝑉′𝑊′𝑋′𝑌′. 𝑉′𝑊′𝑋′𝑌′ is a transformation of 𝑉𝑊𝑋𝑌.

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Describe the single transformation that was used to create 𝑉′𝑊′𝑋′𝑌′. Type of Transformation

Line of reflection is

Down Left

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Reflection

Degrees

Center of rotation is

Rotation

Center, Line, or Movement

Translation

Right

82 | Unit 2

2. The diagram shows 𝑉𝑊𝑋𝑌 and 𝑉′𝑊′𝑋′𝑌′. 𝑉′𝑊′𝑋′𝑌′ is a transformation of 𝑉𝑊𝑋𝑌.

A W'

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Describe the single transformation that was used to create 𝑉′𝑊′𝑋′𝑌′. Type of Transformation

Translation

Center of rotation is

Line of reflection is

Down Left Right

Reflection

Degrees

Rotation

Center, Line, or Movement

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Unit 2 | 83

3. The diagram shows 𝑉𝑊𝑋𝑌 and 𝑉′𝑊′𝑋′𝑌′. 𝑉′𝑊′𝑋′𝑌′ is a transformation of 𝑉𝑊𝑋𝑌.

W B

Describe the single transformation that was used to create 𝑉′𝑊′𝑋′𝑌′.

Reflection Translation

Center, Line, or Movement

Degrees

Center of rotation is

Line of reflection is

Down Left Right

Rotation

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Type of Transformation

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4. Complete the table so the information can be used to distinguish between transformations. First, write your own hints. Then, exchange hints with your partner. Transformation

My Hints

My Partner’s Hints

Reflection

Rotation

Translation

5. Which transformations resulted in congruent figures?

84 | Unit 2

Collaborative Activity: Describing Transformations Each diagram shows ∆𝐹𝐺𝐻 and ∆𝐹′𝐺′𝐻′. Triangle 𝐹′𝐺′𝐻′ is a transformation of ∆𝐹𝐺𝐻. 1. Describe the single transformation that was applied to ∆𝐹𝐺𝐻 to create ∆𝐹′𝐺′𝐻′.

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H' F

G Z

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2. Describe the single transformation that was applied to ∆𝐹𝐺𝐻 to create ∆𝐹′𝐺′𝐻′.

3. The diagram shows ∆𝐹𝐺𝐻 and ∆𝐹′𝐺′𝐻′. ∆𝐹′𝐺′𝐻′ is a transformation of ∆𝐹𝐺𝐻.

G' G'

Describe the single transformation that was applied to ∆𝐹𝐺𝐻 to create ∆𝐹′𝐺′𝐻′.

H' H'

F F''

F Y G Z H

Unit 2 | 85

Lesson Summary The 3 types of rigid motions are reflections, translations, and rotations. Each of these rigid motions can be applied to any figure to create an image that is congruent. A reflection of point 𝐴 across line ℓ can be defined as a transformation that takes point 𝐴 to a point 𝐴′ as described.

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• Point 𝐴′ lies on the line through point 𝐴 that is perpendicular to ℓ, is on the opposite side of line ℓ, and is the same distance from line ℓ as point 𝐴.

• If point 𝐴 happens to be on line ℓ, then 𝐴 and 𝐴′ are both at the same location because they are both a distance of 0 from line ℓ.

A translation slides a figure in a given direction for a given distance with no rotation. The distance and direction is given by a directed line segment.

A directed line segment is a line segment with an arrow at one end specifying a direction.

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The arrow of the directed line segment specifies the direction of the translation, and the length of the directed line segment specifies how far the figure gets translated. A translation can also be described by the direction the figure moves: right, left, up, down. Two descriptions are given of the example shown. • The example shows ∆𝐶𝐷𝐸 translated along directed line segment 𝑣 to ∆𝐶′𝐷′𝐸′.

• The example shows ∆𝐶𝐷𝐸 translated down and to the right to map onto ∆𝐶′𝐷′𝐸′.

86 | Unit 2

Practice Problems

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1. The diagram shows ∆𝐹𝐺𝐻 and ∆𝐹′𝐺′𝐻′. ∆𝐹′𝐺′𝐻′ is a transformation of ∆𝐹𝐺𝐻. Describe the single transformation that was applied to ∆𝐹𝐺𝐻 to create ∆𝐹′𝐺′𝐻′.

2. Lines ℓ and 𝑚 are perpendicular with point of intersection 𝑃.

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Noah says that a 180° rotation, with center 𝑃, has the same effect on points in the plane as reflecting over line 𝑚. Do you agree with Noah? Explain your reasoning.

Unit 2 | 87

3. Match the directed line segment with the image of Polygon 𝑃 being transformed to Polygon 𝑄 by translation by that directed line segment. Transformation

Directed Line Segment

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Translation 1

Translation 2

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Translation 4

Translation 3

88 | Unit 2

Review Problems

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4. What is the measure of ∠𝐴𝐵𝐸?

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5. Lines 𝐴𝐵 and 𝐵𝐶 are perpendicular. The dashed rays bisect ∠𝐴𝐵𝐷 and ∠𝐷𝐵𝐶.

Select all of the statements that must be true.

□ Angle 𝐸𝐵𝐹 is 45°. □ Angle 𝐶𝐵𝐸 is obtuse. □ Angle 𝐶𝐵𝐹 is congruent to ∠𝐷𝐵𝐹. □ Angle 𝐴𝐵𝐶 is congruent to ∠𝐸𝐵𝐹. □ Angle 𝐷𝐵𝐶 is congruent to ∠𝐸𝐵𝐹.

Unit 2 | 89

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Unit 2, Lesson 5: Symmetry – Part 1

Warm-Up: Back to the Start

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Here is a segment 𝐴𝐵:

If you translate the segment up 5 units then down 5 units, it looks the same as it did originally. 1. What other rigid transformations create an image that fits exactly over the original segment?

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2. Are there any single rigid motions that do the same thing?

Exploration Activity: Self Reflection

Determine all the lines of symmetry for the shape your teacher assigns you. Create a visual display about your shape. Include these parts in your display:

• the name of your shape

• the definition of your shape • drawings of each line of symmetry • a description in words of each line of symmetry • one non-example in a different color (a description and drawing of a reflection not over a line of symmetry) © Accelerate Learning Inc. - All Rights Reserved

Unit 2 | 91

Collaborative Activity: Diabolic Diagonals

Lesson Summary This lesson explored symmetry in figures.

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Kiran thinks both diagonals of a kite are lines of symmetry. Tyler thinks only 1 diagonal is a line of symmetry. Who is correct? Explain how you know.

A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is). One special type of symmetry explored is reflectional symmetry.

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A figure has reflectional symmetry if there is a reflection that takes the figure onto itself.

In the case of reflectional symmetry, the line of reflection is called a line of symmetry. A line of symmetry is a line that divides a figure into two congruent parts so that the reflection of either part across the line maps precisely onto the other part.

A regular hexagon has many lines of symmetry. Two examples are shown. What other lines create a reflection where the image is the same as the original figure?

92 | Unit 2

Practice Problems

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1. For each figure, identify any lines of symmetry the figure has.

2. In quadrilateral 𝐵𝐴𝐷𝐶, 𝐴𝐵 = 𝐴𝐷 and 𝐵𝐶 = 𝐷𝐶. The line 𝐴𝐶 is a line of symmetry for this quadrilateral.

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a. Based on the line of symmetry, explain why the diagonals 𝐴𝐶 and 𝐵𝐷 are perpendicular.

b. Based on the line of symmetry, explain why angles 𝐴𝐵𝐶 and 𝐴𝐷𝐶 have the same measure.

Unit 2 | 93

Review Problems

4. Lines ℓ and 𝑚 are perpendicular.

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3. Three line segments form the letter Z. Rotate the letter Z counterclockwise around the midpoint of segment 𝐵𝐶 by 180°. Describe the result.

𝑚⊥ℓ

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□ A □ B □ C □ D □ E

Sometimes reflecting a point over 𝑚 has the same effect as rotating the point 180° using center 𝑃. Select all labeled points which have the same image for both transformations.

94 | Unit 2

Unit 2, Lesson 6: Symmetry – Part 2

Warm-Up: Which One Doesn’t Belong: Symmetry Which one doesn’t belong? B

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Exploration Activity: Self-Rotation

Determine all the angles of rotation that create symmetry for the shape your teacher assigns you. Create a visual display about your shape. Include these parts in your display:

• the name of your shape

• the definition of your shape • drawings of each rotation that creates symmetry • a description in words of each rotation that creates symmetry, including the center, angle, and direction of rotation • one non-example (a description and drawing of a rotation that does not result in symmetry) © Accelerate Learning Inc. - All Rights Reserved

Unit 2 | 95

Collaborative Activity: Parallelogram Symmetry Clare says, “Last class I thought the parallelogram would have reflection symmetry. I tried using a diagonal as the line of symmetry, but it didn’t work. So now I’m doubting that it has rotation symmetry.” Lin says, “I thought that too at first, but now I think that a parallelogram does have rotation symmetry. Here, look at this.”

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How could Lin describe to Clare the symmetry she sees?

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Guided Activity: Point Symmetry

Some polygons that have rotational symmetry also have point symmetry. The table shows examples of polygons with and without point symmetry. Polygons without Point Symmetry

Polygons with Point Symmetry

96 | Unit 2

1. What do you notice about the polygons with point symmetry compared to the polygons without point symmetry?

2. Complete the table by listing the angle(s) of rotation that produce(s) a symmetrical polygon. Figure

Angle(s)

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Polygon

Rectangle

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Octagon

Hexagon

Pentagon

Triangle

Heptagon

Unit 2 | 97

3. Discuss with your partner the similarities and differences of the angles of rotation for the polygons with point symmetry compared to the polygons without point symmetry.

4. Complete the statement. 90° 180° 360°

about its center takes

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A polygon has point symmetry if rotating the figure the shape onto itself.

Lesson Summary

A figure has rotational symmetry if there is a rotation between 0° and 360° that takes the figure onto itself.

A figure has rotational symmetry if there is a rotation that takes the figure onto itself. (We don’t count rotations using angles such as 0° and 360° that leave every point on the figure where it is).

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A figure has point symmetry if a rotation of 180° takes the figure onto itself. The regular hexagon shown has both rotational and point symmetry.

A figure is said to have self-congruence if rigid motions can take the shape onto itself.

98 | Unit 2

Practice Problems

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1. For each figure, identify any angles of rotation that create symmetry.

2. A triangle has rotation symmetry that can take any of its vertices to any of its other vertices. Select all conclusions that we can reach from this.

□ All sides of the triangle have the same length. □ All angles of the triangle have the same measure. □ All rotations take one half of the triangle to the other half of the triangle.

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3. Circle all of the letters that have point symmetry.

E S

M O

Z H Unit 2 | 99

4. Select all the angles of rotation that produce symmetry for this flower.

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□ 45° □ 90° □ 135° □ 180° □ 225° □ 270°

Review Problems

5. A triangle has a line of symmetry. Select all conclusions that must be true.

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□ All sides of the triangle have the same length. □ All angles of the triangle have the same measure. □ No sides of the triangle have the same length. □ No angles of the triangle have the same measure. □ Two sides of the triangle have the same length. □ Two angles of the triangle have the same measure.

6. Here are 4 triangles that have each been transformed by a different transformation. Which transformation is not a rigid transformation?

100 | Unit 2

Unit 2, Lesson 7: Working with Rigid Transformations

Warm-Up: Math Talk: From Here to There

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Segment 𝐶𝐷 is the perpendicular bisector of segment 𝐴𝐵. Find each transformation mentally.

• A transformation that takes 𝐴 to 𝐵.

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• A transformation that takes 𝐵 to 𝐴.

• A transformation that takes 𝐶 to 𝐷.

• A transformation that takes 𝐷 to 𝐶.

Unit 2 | 101

Exploration Activity: Translate, Rotate, Reflect

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This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

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Mai suspects triangle 𝐴𝐵𝐶 is congruent to triangle 𝐷𝐸𝐹. She thinks these steps will work to show there is a rigid transformation from 𝐴𝐵𝐶 to 𝐷𝐸𝐹. • Translate by directed line segment 𝑣. • Rotate the image

degrees clockwise around point 𝐷.

• Reflect that image over line 𝐷𝐸.

Draw each image and determine the angle of rotation needed for these steps to take 𝐴𝐵𝐶 to 𝐷𝐸𝐹.

102 | Unit 2

Collaborative Activity: How Did This Get There? 1. Your teacher will give you a set of cards that show transformations of figures. a. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.

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b. Then sort the cards into categories in a different way. Be prepared to explain the meaning of your new categories.

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2. For each card with a rigid transformation: write a sequence of rotations, translations, and reflections to get from the original figure to the image. Be precise.

Unit 2 | 103

Guided Activity: Reflecting on Reflection

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This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

Diego says, ”I see why a reflection could take 𝑅𝑆𝑇𝑈 to 𝑅′𝑆′𝑇′𝑈′, but I’m not sure where the line of reflection is. I’ll just guess.”

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1. How could Diego see that a reflection could work without knowing where the line of reflection is?

2. How could Diego find an exact line of reflection that would work?

104 | Unit 2

Lesson Summary If 2 figures are congruent, a rigid transformation can always take one onto the other.

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Congruent triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are shown. It looks like ∆𝐷𝐸𝐹 might be a reflection and translation of ∆𝐴𝐵𝐶. But is there a way to describe a sequence of transformations without guessing where the line of reflection might be?

The goal is to take the image of 𝐸 onto 𝐵. Then, take the image of 𝐷 onto 𝐴 without moving 𝐸 and 𝐵. Finally, take the image of 𝐹 onto 𝐶 without moving any of the corresponding points.

• This sequence can start with translation. Translate ∆𝐷𝐸𝐹 by the directed line segment from 𝐸 to 𝐵, as shown.

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• Now, a pair of corresponding points coincides. From here, is there a transformation that could be used to take 𝐷′ onto 𝐴 that leaves 𝐵 and 𝐸′ in place? Rotations have a fixed point, so rotate ∆𝐷′𝐸′𝐹′ by ∠𝐷′𝐵𝐴, using point 𝐵 as the center, as shown.

• At this point, 2 pairs of corresponding points coincide. Reflecting across 𝐴𝐵 will take ∆𝐷″𝐸″𝐹″ onto ∆𝐴𝐵𝐶. The points 𝐷″ and 𝐸″ won’t move, since points on the line of reflection don’t move. How is it known that 𝐹″ will end up on 𝐶? Since the triangles are congruent, 𝐹″ and 𝐶 are the same distance from the line of reflection. It’s always possible to describe transformations using existing points, angles, and segments.

Unit 2 | 105

Practice Problems

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1. Quadrilateral 𝐴𝐵𝐶𝐷 is congruent to quadrilateral 𝐴′𝐵′𝐶′𝐷′. Describe a sequence of rigid motions that takes 𝐴 to 𝐴′, 𝐵 to 𝐵′, 𝐶 to 𝐶′, and 𝐷 to 𝐷′.

2. Select all transformations that must take any point 𝐴 to any point 𝐵.

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□ Rotation of 180° around 𝐴 □ Rotation of 180° around 𝐵 □ Rotation of 180° around the midpoint of segment 𝐴𝐵 □ Reflection across the line 𝐴𝐵 □ Reflection across the perpendicular bisector of segment 𝐴𝐵 □ Translation by the directed line segment 𝐴𝐵 □ Translation by the directed line segment 𝐵𝐴

3. Triangle 𝐴𝐵𝐶 is congruent to triangle 𝐴′𝐵′𝐶′. Describe a sequence of rigid motions that takes 𝐴 to 𝐴′, 𝐵 to 𝐵′, and 𝐶 to 𝐶′.

106 | Unit 2

Review Problems 4. A triangle has rotation symmetry that can take any of its vertices to any of its other vertices. Select all conclusions that we can reach from this.

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□ All sides of the triangle have the same length. □ All angles of the triangle have the same measure. □ All rotations take one half of the triangle to the other half of the triangle. □ It is a right triangle. □ None of the sides of the triangle have the same length. □ None of the angles of the triangle have the same measure.

5. A right triangle has a line of symmetry. Select all conclusions that must be true.

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□ All sides of the triangle have the same length. □ All angles of the triangle have the same measure. □ Two sides of the triangle have the same length. □ Two angles of the triangle have the same measure. □ No sides of the triangle have the same length. □ No angles of the triangle have the same measure.

6. In quadrilateral 𝐵𝐴𝐷𝐶, 𝐴𝐵 = 𝐴𝐷 and 𝐵𝐶 = 𝐷𝐶. The line 𝐴𝐶 is a line of symmetry for this quadrilateral. Based on the line of symmetry, explain why angles 𝐴𝐶𝐵 and 𝐴𝐶𝐷 have the same measure.

Unit 2 | 107

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Unit 2, Lesson 8: Rigid Transformations on the Coordinate Plane

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Warm-Up: Traversing the Plane

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1. How far is point 𝐴 from point 𝐵?

2. What transformations will take point 𝐴 to point 𝐵?

Unit 2 | 109

Exploration Activity: Transforming with Coordinates

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First, predict where each transformation will land. Next, carry out the transformation.

1. Rotate Figure 𝐻 clockwise using center (2, 0) by 90°. Translate the image by the directed line segment from (2, 0) to (3, −4). Label the result 𝑅.

2. Reflect Figure 𝐻 across the 𝑦-axis. Rotate the image counterclockwise using center (0, 0) by 90°. Label the result 𝐿.

110 | Unit 2

Collaborative Activity: Describing Sequences on the Coordinate Plane

Describe a sequence of rigid motions that takes ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹.

2. Polygon 𝐾𝑀𝑁𝑃 is the image of polygon 𝐴𝐵𝐶𝐷 after a sequence of transformations, as shown.

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1. Triangle 𝐷𝐸𝐹 is the image of ∆𝐴𝐵𝐶 after a sequence of transformations, as shown.

cr -6 -6

--5 5 -4 -4 N

-3 3

1 K 2 -2

1 O -1 -1 1 P -2 -2 -3 -3

7 x

--4 4 5 --5 -6 -6

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Describe a sequence of rigid motions that takes 𝐴𝐵𝐶𝐷 onto 𝐾𝑀𝑁𝑃.

Unit 2 | 111

3. Triangle 𝑁𝐺𝑀 is the image of ∆𝑋𝐶𝐻 after a sequence of transformations, as shown. Describe a sequence of transformations that takes ∆𝑋𝐶𝐻 onto ∆𝑁𝐺𝑀.

y N

8 7 6 5

3 2

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-7 7 -6 -6 -5 -5 --4 4 -3 3 -2 2 -1 1 O 1 --1

-2 2 -3 3

-4 -4

-5 -5

Lesson Summary

-6 -6

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Sequences of transformations were explored off of the coordinate plane earlier in this unit, but sequences of transformations can be performed on the coordinate plane as well. When transformations are performed on the coordinate plane, ordered pairs and locations on the coordinate plane can be used to describe the transformations. There is a sequence of rigid motions that can take ∆𝐴𝐵𝐶 to ∆𝐷𝐸𝐹. One possible sequence is described. • First, reflect ∆𝐴𝐵𝐶 across the 𝑦-axis.

• Then, translate the image left 2 units.

112 | Unit 2

Practice Problems

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1. Reflect triangle 𝐴𝐵𝐶 over the line 𝑥 = −3. Translate the image by the directed line segment from (0, 0) to (4, 1).

What are the coordinates of the vertices in the final image?

Unit 2 | 113

2. Triangle 𝐴𝐵𝐶 has coordinates 𝐴 = (1, 3), 𝐵 = (2, 0), and 𝐶 = (4, 1). The image of this triangle after a sequence of transformations is triangle 𝐴′𝐵′𝐶′ where 𝐴′ = (−5, −3), 𝐵′ = (−4, 0), and 𝐶′ = (−2, −1).

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Write a sequence of transformations that takes triangle 𝐴𝐵𝐶 to triangle 𝐴′𝐵′𝐶′.

Review Problems

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3. Three line segments form the letter N. Rotate the letter N counterclockwise around the midpoint of segment 𝐵𝐶 by 180°. Describe the result.

4. Select all the angles of rotation that produce symmetry for this flower.

□ 30° □ 45° □ 60° □ 90° □ 120° □ 135° □ 180°

114 | Unit 2

Unit 2, Lesson 9: Describing Transformations on the Coordinate Plane

Warm-Up: Math Talk: Transforming a Point

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Mentally find the coordinates of the image of 𝐴 under each transformation.

• Translate 𝐴 by the directed line segment from (0,0) to (0,2).

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• Translate 𝐴 by the directed line segment from (0,0) to (−4,0). • Reflect 𝐴 across the 𝑥-axis.

• Rotate 𝐴 180 degrees clockwise using the origin as a center.

Unit 2 | 115

Exploration Activity: Transformations Using Coordinates

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A coordinate plane is shown.

1. For each point (𝑥, 𝑦), find its image under the transformation (𝑥 + 12, 𝑦 − 2). a. 𝐴 = (−10, 5)

b. 𝐵 = (−4, 9) c. 𝐶 = (−2, 6)

2. Sketch ∆𝐴𝐵𝐶 and its image on the grid.

3. Describe the transformation (𝑥, 𝑦) → (𝑥 + 12, 𝑦 − 2) in words.

116 | Unit 2

4. The vertices of a quadrilateral are given in the table. a. For each given point (𝑥, 𝑦), find (−𝑥, 𝑦) and (𝑥, −𝑦). (𝒙, 𝒚)

(−𝒙, 𝒚)

(−1, −3) (−1, 1)

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(5, 1)

(𝒙, −𝒚)

(5, −3)

b. Sketch all 3 figures with the vertices determined in question 4 on the coordinate plane.

c. Describe the transformation (𝑥, 𝑦) → (−𝑥, 𝑦) in words.

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d. Describe the transformation (𝑥, 𝑦) → (𝑥, −𝑦) in words.

Unit 2 | 117

Collaborative Activity: What Does It Do? Quadrilateral 𝐴𝐵𝐶𝐷 is shown on the coordinate plane. 10

9 8 7

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6 5 4 3 2 1 -10 -9

-8

-7

-6

-5

-4

-3

-2

-1

0 -1 -2

-3

-4 -5 -6

-7 -8

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-9

-10

1. Apply each rule to quadrilateral 𝐴𝐵𝐶𝐷, and graph the resulting image. Label the image with the letter indicated, and then describe the transformation.

a. Figure 𝑄: (𝑥, 𝑦) → (−𝑥, −𝑦) b. Figure 𝑅: (𝑥, 𝑦) → (−𝑦, −𝑥) c. Figure 𝑆: (𝑥, 𝑦) → (𝑦, −𝑥)

118 | Unit 2

Lesson Summary Transformations on the coordinate plane can be thought of as functions that take points on the plane as inputs and give other points as outputs. The algebraic descriptions of rigid transformations are shown in the table. Translations Left

(𝑥, 𝑦) → (𝑥 + 𝑎, 𝑦)

(𝑥, 𝑦) → (𝑥 − 𝑎, 𝑦)

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Right

Down

(𝑥, 𝑦) → (𝑥, 𝑦 + 𝑏)

(𝑥, 𝑦) → (𝑥 , 𝑦 − 𝑏)

Rotations Clockwise about the Origin

Rotations of 𝟗𝟎°

Rotations of 𝟏𝟖𝟎°

(𝑥, 𝑦) → (𝑦, −𝑥)

(𝑥, 𝑦) → (−𝑥, −𝑦)

Rotations of 𝟐𝟕𝟎° (𝑥, 𝑦) → (−𝑦, 𝑥)

(𝑥, 𝑦) → (−𝑥, −𝑦) Reflections

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(𝑥, 𝑦) → (−𝑦, 𝑥)

Rotations of 𝟏𝟖𝟎°

Rotations of 𝟗𝟎°

Rotations Counterclockwise about the Origin

Rotations of 𝟐𝟕𝟎° (𝑥, 𝑦) → (𝑦, −𝑥)

Across the 𝒙-axis

Across the 𝒚-axis

Across 𝒚 = 𝒙

Across 𝒚 = −𝒙

(𝑥, 𝑦) → (𝑥, −𝑦)

(𝑥, 𝑦) → (−𝑦, −𝑥)

(𝑥, 𝑦) → (𝑦, 𝑥)

(𝑥, 𝑦) → (−𝑥, 𝑦)

Unit 2 | 119

Practice Problems 1. Match each coordinate rule to the description of its resulting transformation.

B. (𝑥, 𝑦) → (𝑦, 𝑥) C. (𝑥, 𝑦) → (𝑥, 𝑦 + 4) D. (𝑥, 𝑦) → (−𝑥, −𝑦)

E. (𝑥, 𝑦) → (𝑥 − 3, 𝑦 + 4)

ii. Translate by the directed line segment from (0, 0) to (3, 0). iii. Reflect across the line 𝑦 = 𝑥.

iv. Rotate 180° about the origin.

v. Translate up 4 units and left 3 units.

i. Translate by the directed line segment from (0, 0) to (0, 4).

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A. (𝑥, 𝑦) → (𝑥 + 3, 𝑦)

a. Draw the image of triangle 𝐴𝐵𝐶 under the transformation (𝑥, 𝑦) → (𝑥 − 4, 𝑦 + 1). Label the result 𝑇.

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b. Draw the image of triangle 𝐴𝐵𝐶 under the transformation (𝑥, 𝑦) → (−𝑥, 𝑦). Label the result 𝑅.

120 | Unit 2

3. Some transformation rules are shown. For each rule, describe the transformation in words. a. (𝑥, 𝑦) → (𝑥 − 2, 𝑦 − 3)

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b. (𝑥, 𝑦) → (−𝑥, 𝑦) c. (𝑥, 𝑦) → ( −𝑦, 𝑥) d. (𝑥, 𝑦) → (2 − 𝑥, 𝑦)

Review Problems

4. Reflect ∆𝐴𝐵𝐶 over the line 𝑥 = 0 to create ∆𝐴′𝐵′𝐶′. Then, reflect ∆𝐴′𝐵′𝐶′ over the line 𝑦 = 0 to create ∆𝐴″𝐵″𝐶″.

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Which single transformation takes ∆𝐴𝐵𝐶 to ∆𝐴″𝐵″𝐶″?

A. Translate ∆𝐴𝐵𝐶 by the directed line segment from (1, 1) to (−2, 1).

B. Reflect ∆𝐴𝐵𝐶 across the line 𝑦 = −𝑥.

C. Rotate ∆𝐴𝐵𝐶 counterclockwise 180°, using the origin as the center. D. Rotate ∆𝐴𝐵𝐶 clockwise 90°, using the origin as the center.

Unit 2 | 121

5. Reflect ∆𝐴𝐵𝐶 over the line 𝑦 = 2. Translate the image by the directed line segment from (0, 0) to (3, 2).

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What are the coordinates of the vertices in the final image?

122 | Unit 2

Unit 2, Lesson 10: Sequences of Transformations

Warm-Up: Notice and Wonder: Obstacles

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What do you notice? What do you wonder?

Unit 2 | 123

Guided Activity: Sequences of Transformations 1. Quadrilateral 𝑅𝐾𝐻𝐷 was transformed to create quadrilateral 𝑅′𝐾′𝐻′𝐷′. Then, quadrilateral 𝑅′𝐾′𝐻′𝐷′ was transformed to create quadrilateral 𝑅′′𝐾′′𝐻′′𝐷′′. The quadrilaterals are shown on the coordinate plane.

c. Complete the table.

-6 -5 -4 -3 -2 -1 0 -1 H -2 K -3 R -4

b. On the graph, show the mappings of the vertices of the second transformation.

Quadrilateral 𝑹𝑲𝑯𝑫 Transformed to Quadrilateral 𝑹′𝑲′𝑯′𝑫′

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D'' D''

D' D'

H'''' H K'' K''

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a. On the graph, show the mappings of the vertices of the first transformation.

7 6 5 4 3 2 1

R'' R'' 1 2 3 4 5 6 7 8 9 10 H' H' K' K' R' R'

Quadrilateral 𝑹′𝑲′𝑯′𝑫′ Transformed to Quadrilateral 𝑹′′𝑲′′𝑯′′𝑫′′

Algebraic Description

d. Explain how to determine if a sequence of transformations preserves distance.

124 | Unit 2

2. Triangles 𝐻𝑇𝐶, 𝑉𝐾𝐶, and 𝑊𝐺𝑌 are shown on the coordinate plane. Triangle 𝐻𝑇𝐶 is the preimage. Triangle 𝑉𝐾𝐶 is the result of the first transformation, and ∆𝑊𝐺𝑌 is the result of the second transformation.

Y 30

a. Describe the first transformation.

20 15 10

G K

-30 -25 -20 -15 -10 -5 0 -5

5 10 15 20 25 30 35

-10

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-15

b. Use tracing paper to show that the transformation preserved distance. c. Describe the second transformation.

d. Use tracing paper to show that the second transformation preserved distance.

center of rotation line of reflection

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Because the

e. Work with your partner to complete the statement.

an algebraic description

can cannot

∆𝐻𝑇𝐶 ∆𝑉𝐾𝐶

be used.

(0, 0), (−5, −10), 𝑦 = 0, 𝑦 = 10,

f. When Judeline and Carmen completed the statement above, they had different correct answers. Write the second correct answer.

Unit 2 | 125

a. Explain which features of the two triangles can be used to determine that the transformations were rigid motions.

4 3 2

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1

-2 -3

G'''' B 1

V'' V''

N''''

-4

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3. Triangles 𝑁𝑉𝐺 and 𝑁′′𝑉′′𝐺′′ are shown on the coordinate plane. Triangle 𝑁𝑉𝐺 was transformed to create ∆𝑁′′𝑉′′𝐺′′.

-5 -6 -7

b. Adele told Kiriam that she knows one of the rigid motions is a rotation. Discuss with your partner how Adele may have drawn her conclusion based on clues in the location of the vertices. Write a summary of your discussion.

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c. Work with your partner to determine a sequence of transformations that includes a rotation about the point 𝐵 so that, when applied, ∆𝑁𝑉𝐺 will map onto ∆𝑁′′𝑉′′𝐺′′. Complete the table. Transformation

Mapping

Algebraic Description

∆𝑁𝑉𝐺 → ∆𝑁′𝑉′𝐺′

126 | Unit 2

∆𝑁′𝑉′𝐺′ → ∆𝑁′′𝑉′′𝐺′′

Collaborative Activity: Sequences of Transformations 1. Triangles 𝑇𝑊𝐹 and 𝑇′′𝑊′′𝐹′′ are shown on the coordinate plane. 9 8

F'' F''

7 6 4 3 2

W -10

-12

-8

-6

-4

-2

1 0 -1 -2 -3

T'' T''

-4

W W''''

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-5 -6

-7

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Determine a sequence of transformations that will map ∆𝑇𝑊𝐹 onto ∆𝑇′′𝑊′′𝐹′′. Complete the table.

Transformation

Mapping

Algebraic Description

∆𝑇𝑊𝐹 → ∆𝑇′𝑊′𝐹′

∆𝑇′𝑊′𝐹′ → ∆𝑇′′𝑊′′𝐹′′

Unit 2 | 127

Lesson Summary This lesson explored sequences of transformations on the coordinate plane and how to describe them verbally and algebraically. A sequence of transformations is a set of translations, rotations, reflections, and dilations on a figure. The transformations are performed in a given order.

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Algebraic descriptions can be used to describe translations, reflections, and rotations in a sequence on a coordinate plane. • To use an algebraic description of a rotation on the coordinate plane, the center of rotation must be the origin, (0, 0). • To use an algebraic description of a reflection on the coordinate plane, the line of reflection must be the 𝑥-axis (𝑦 = 0), the 𝑦-axis (𝑥 = 0), 𝑦 = 𝑥, or 𝑦 = −𝑥.

Practice Problems 1. Quadrilateral 𝑀𝐶𝐺𝑊 was transformed to create quadrilateral 𝑀′′𝐶′′𝐺′′𝑊′′.

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a. Describe the sequence of transformations that will map quadrilateral 𝑀𝐶𝐺𝑊 onto quadrilateral 𝑀′′𝐶′′𝐺′′𝑊′′.

M 128 | Unit 2

W M

4 2

M''''

x -10

-8

-6

-4

-2 C C''''

0 -2 -4

-6

W'' W''

b. Write the sequence of transformations using algebraic descriptions.

-8 G''''

-10 -12

2. Triangle 𝐹𝐻𝑀 was transformed to create ∆𝐹′′𝐻′′𝑀′′, as shown. 6

y H"

5 4 3

F" M"

2 1

x 1

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-7 -6 -5 -4 -3 -2 -1 0 H -1 -2

-3 -4 -5

Complete the table by determining a sequence of transformations that will map ∆𝐹𝐻𝑀 onto ∆𝐹′′𝐻′′𝑀′′.

Mapping

Algebraic Description

Transformation

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∆𝐹𝐻𝑀 → ∆𝐹′𝐻′𝑀′

∆𝐹′𝐻′𝑀′ → ∆𝐹′′𝐻′′𝑀′′

Unit 2 | 129

Review Problems 3. Reflect triangle 𝐴𝐵𝐶 over the line 𝑥 = −2. Call this new triangle 𝐴′𝐵′𝐶′. Then reflect triangle 𝐴′𝐵′𝐶′ over the line 𝑥 = 0. Call the resulting triangle 𝐴″𝐵″𝐶″.

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Describe a single transformation that takes 𝐴𝐵𝐶 to 𝐴″𝐵″𝐶″.

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4. The figures are congruent. Select all the sequences of transformations that would take Figure 1 to Figure 2.

□ Translate by directed line segment 𝐴𝐷. □ Rotate 180° around point 𝐸. □ Translate by directed line segment 𝐴𝐸 and reflect across 𝐴𝐶.

□ Translate by directed line segment 𝐶𝐸 and

rotate 90° counterclockwise around point 𝐸.

□ Rotate 180° around point 𝐶, translate by

directed line segment 𝐶𝐸, and reflect across segment 𝐸𝐹.

□ Reflect across segment 𝐴𝐵, rotate clockwise

by angle 𝐸𝐹𝐵 using center 𝐹, then reflect across segment 𝐸𝐹.

130 | Unit 2

Unit 2, Lesson 11: Applying a Sequence of Transformations

Warm-Up: Notice and Wonder

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What do you notice? What do you wonder?

Unit 2 | 131

Collaborative Activity: Applying Sequences of Transformations from Written Descriptions

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1. Quadrilateral 𝐶𝐻𝑀𝑊 is shown on the coordinate plane.

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a. Draw the quadrilateral that is the result of reflecting across the 𝑦-axis and then translating left 3 units and up 1 unit. Label the new quadrilateral 𝐵𝑉𝑋𝑁. b. Compare your graph with your partner’s. Make any necessary adjustments.

c. Work with your partner to explain why quadrilateral 𝐵𝑉𝑋𝑁 is a result of a sequence of transformations.

132 | Unit 2

2. Quadrilateral 𝐶𝐻𝑀𝑊 is shown on the coordinate plane.

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a. Draw the quadrilateral that is the result of rotating 180° counterclockwise about the origin and then reflecting across the 𝑥-axis. Label the new quadrilateral 𝐾𝐷𝑃𝐿.

3. Quadrilateral 𝐶𝐻𝑀𝑊 is shown on the coordinate plane.

b. Compare your graph with your partner. Make any necessary adjustments.

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a. Draw the quadrilateral that is the result of rotating 90° clockwise about point 𝑊 and then translating left 2 and up 5. Label the new quadrilateral 𝑅𝑌𝐹𝐺.

b. Compare your graph with your partner’s. Make any necessary adjustments.

Unit 2 | 133

Collaborative Activity: Applying Sequence of Transformations from Algebraic Descriptions 1. Quadrilateral 𝑅𝐷𝑌𝑊 is shown on the coordinate plane. 8 6 5 4 3 2 1 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 -2 -3

-5

9 10 11 12 13

-6

-4

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-7 -8

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-9

-10 -11

Perform the sequence of transformations to quadrilateral 𝑅𝐷𝑌𝑊. Assume all rotations are centered at the origin. Draw the result of each transformation on the graph. • (𝑥, 𝑦) → (−𝑥, 𝑦) Label as 𝑅′𝐷′𝑌′𝑊′.

• (𝑥, 𝑦) → (−𝑦, 𝑥) Label as 𝑅′′𝐷′′𝑌′′𝑊′′.

134 | Unit 2

2. Quadrilateral 𝐵𝑋𝐻𝐾 is shown on the coordinate plane. Perform the sequence of transformations to quadrilateral 𝐵𝑋𝐻𝐾. Assume all rotations are centered at the origin. Draw the result of each transformation on the graph.

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• (𝑥, 𝑦) → (𝑥 + 10, 𝑦 − 4) Label as 𝐵′𝑋′𝐻′𝐾′. • (𝑥, 𝑦) → (𝑥, −𝑦) Label as 𝐵′′𝑋′′𝐻′′𝐾′′.

3. Triangle 𝐻𝐹𝑋 is shown on the coordinate plane. 25

-25

-20

-15

-10

-5

-10 -15 -20 -25

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-30

Perform the sequence of transformations on ∆𝐻𝐹𝑋. Assume all rotations are centered at the origin. Label the result of all three transformations as ∆𝑇𝑀𝑅. • (𝑥, 𝑦) → (𝑥 − 22, 𝑦 + 9) • (𝑥, 𝑦) → (−𝑥, 𝑦)

• (𝑥, 𝑦) → (−𝑥, −𝑦)

Unit 2 | 135

Lesson Summary The effects of transformations on coordinates are summarized in the tables. Rotations Clockwise about the Origin Rotations of 𝟗𝟎°

Rotations of 𝟏𝟖𝟎°

Change the 𝑦-coordinate Change both coordinates to to its opposite, and then their opposites. switch the coordinates.

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Change the 𝑥-coordinate to its opposite, and then switch the coordinates.

Rotations of 𝟐𝟕𝟎°

Rotations Counterclockwise about the Origin Rotations of 𝟗𝟎°

Change the 𝑦-coordinate to its opposite, and then switch the coordinates.

Rotations of 𝟏𝟖𝟎°

Change both coordinates to their opposites.

Rotations of 𝟐𝟕𝟎°

Change the 𝑥-coordinate to its opposite, and then switch the coordinates.

Right

Add to the 𝑥-coordinate.

Translations

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Add to the 𝑦-coordinate. Across the 𝒙-axis

Reflections

Left

Subtract from the 𝑥-coordinate. Down

Subtract from the 𝑦-coordinate. Across the 𝒚-axis

The 𝑦-coordinate changes to its opposite.

The 𝑥-coordinate changes to its opposite.

The coordinates switch.

The coordinates switch and become their opposites.

Across 𝒚 = 𝒙

136 | Unit 2

Across 𝒚 = −𝒙

Practice Problems 1. Polygon 𝐵𝐾𝐻𝑊𝑌 is shown on the coordinate plane.

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On the coordinate plane, translate polygon 𝐵𝐾𝐻𝑊𝑌 right 5 and down 3. Then, reflect the result across the line 𝑥 = 4. Label the resulting polygon 𝐵′′𝐾′′𝐻′′𝑊′′𝑌′′.

2. Quadrilateral 𝑋𝐻𝑇𝑀 is shown on the coordinate plane.

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Perform the sequence of transformations on quadrilateral 𝑋𝐻𝑇𝑀. Assume all rotations are centered at the origin. Label the result of the transformations as quadrilateral 𝑊𝑄𝑃𝑍.

7 6 5 4 3 2 1

-8 -7 -6 -5 -4 -3 -2 -1 0 -1

• (𝑥, 𝑦) → (−𝑥, 𝑦)

• (𝑥, 𝑦) → (−𝑦, 𝑥)

• (𝑥, 𝑦) → (𝑥 + 4, 𝑦 − 1)

-2

9 10 11 12 13

-3

-4 -5 -6 -7 -8

-9

Unit 2 | 137

3. Triangle 𝐻𝑀𝑇 is shown on the coordinate plane.

9 8 7

Draw ∆𝐵𝑊𝐹, which is the result of translating ∆𝐻𝑀𝑇 left 3 and up 3 and then rotating the result 90° clockwise about the origin.

6 5 4 3 2

1 1

9 10 11 12 13 14 15

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-6 -5 -4 -3 -2 -1 0 -1 -2 -3

-4 -5 -6 -7

-8

Review Problem

4. Lines 𝐴𝐷 and 𝐸𝐶 meet at point 𝐵.

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Which of these must be true? Select all that apply.

□ A 180° clockwise rotation using center 𝐵 takes 𝐷 to 𝐴.

□ The image of 𝐷 after a 180° rotation using center 𝐵 lies on ray 𝐵𝐴.

□ If a 180° rotation using center 𝐵 takes 𝐶 to 𝐸

then it also takes 𝐸 to 𝐶.

□ Angle 𝐴𝐵𝐶 is congruent to angle 𝐷𝐵𝐸. □ Angle 𝐴𝐵𝐸 is congruent to angle 𝐴𝐵𝐶.

138 | Unit 2

Unit 2, Lesson 12: Parallel Lines and Transversals

Warm-Up: Supplementary Angles Mentally evaluate all of the missing angle measures in each figure. Figure B

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Figure A

Figure D

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Figure C

Unit 2 | 139

Collaborative Activity: Revisiting Angle Relationships 1. In each diagram, 𝑇𝑀 ∥ 𝑁𝑊, and 𝑌𝐿 is a transversal.

a. Determine whether the highlighted angles are congruent or supplementary.

Congruent

Diagram B

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Diagram A

Supplementary

Diagram D

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Diagram C

Congruent

Supplementary

Congruent

Supplementary

Congruent

140 | Unit 2

b. Use the word bank to complete the statements. Word Bank alternate interior

alternate exterior

corresponding

consecutive interior angles.

The marked angles in diagram B are

angles.

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The marked angles in diagram A are

The marked angles in diagram C are

angles.

The marked angles in diagram D are

angles.

The consecutive exterior angle theorem states that if 2 parallel lines are intersected by a transversal, then exterior angles on the same side of the transversal are supplementary.

c. Identify the 2 pairs of consecutive exterior angles.

Guided Activity: Angles in Parallel Lines

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1. Line 𝐻𝐿 is parallel to 𝑅𝐺, 𝐶𝑉 and 𝐾𝐷 are transversals, 𝑚∠𝐾𝐵𝑀 = 117°, and 𝑚∠𝑉𝑇𝑆 = 134°. Determine the measurement of each angle.

a. 𝑚∠𝐷𝑆𝐺 =

b. 𝑚∠𝐿𝑀𝐶 = c. 𝑚∠𝑅𝑇𝑉 =

d. 𝑚∠𝐵𝑊𝑀 =

Unit 2 | 141

2. In the figure shown, 𝐵𝑉 ∥ 𝑇𝐻, 𝑚∠𝐺𝑋𝐻 = (5(2𝑥 − 1))°, 𝑚∠𝐺𝑀𝑁 = (4𝑥 − 11)°, 𝑚∠𝑋𝑃𝐺 = (3𝑦 − 8)°, and 𝑚∠𝑉𝑁𝐺 = (7𝑦 − 52)°.

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a. Find the value of 𝑥.

b. Find the value of 𝑦.

Collaborative Activity: Solving Problems Using Angle Relationships, Postulates, and Theorems

With your partner, sort the cards to match each image with the type of angle pair represented and the value of 𝑥. Record your answers in the table, showing your work in the last column.

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Letter of Image

Work and Answer

Type of Angle Pair

142 | Unit 2

Letter of Image

Work and Answer

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Type of Angle Pair

Unit 2 | 143

Letter of Image

Work and Answer

Lesson Summary

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Type of Angle Pair

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Relationships between angles formed when parallel lines are crossed by a transversal can be used to determine unknown angle measures. • The alternate interior angle theorem states that if 2 parallel lines are intersected by a transversal, then alternate interior angles are congruent. • The alternate exterior angle theorem states that if 2 parallel lines are intersected by a transversal, then alternate exterior angles are congruent.

• The corresponding angles theorem states that if 2 parallel lines are intersected by a transversal, then corresponding angles are congruent. • The consecutive interior angle theorem states that if 2 parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary. • The consecutive exterior angle theorem states that if 2 parallel lines are intersected by a transversal, then exterior angles on the same side of the transversal are supplementary. 144 | Unit 2

Practice Problems 1. In the figure shown, 𝐹𝑌 ∥ 𝐻𝑇. Lines 𝐵𝐺 and 𝐾𝑃 are transversals, 𝑚∠𝐾𝑋𝑌 = (15𝑥 − 2)°, 𝑚∠𝑇𝑊𝑃 = (7𝑥 + 6)°, 𝑚∠𝑀𝑅𝐻 = (6𝑦 + 4)°, 𝑚∠𝐹𝑀𝑅 = (9𝑦 − 11.5)°. Determine the measurement of each angle. Show your work.

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a. 𝑚∠𝐾𝑋𝑌 =

b. 𝑚∠𝑇𝑊𝑃 =

c. 𝑚∠𝑀𝑅𝐻 = d. 𝑚∠𝐹𝑀𝑅 =

e. 𝑚∠𝐵𝑀𝐹 =

f. 𝑚∠𝑋𝑊𝑇 =

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2. Lines 𝑃𝑄 and 𝑅𝑆 are parallel, and 𝐴𝐷 is a transversal. a. Determine the value of 𝑛.

b. What is 𝑚∠𝐴𝐵𝑄? c. What is 𝑚∠𝐵𝐶𝑅?

Unit 2 | 145

3. Lines 𝐹𝐺 and 𝐽𝐾 are parallel, and 𝑀𝑇 is a transversal. a. Determine the value of 𝑝.

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b. What is 𝑚∠𝐹𝑁𝑇?

c. What is 𝑚∠𝐾𝑇𝑈?

Review Problems

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4. Triangle 𝐴𝐵𝐶 is congruent to ∆𝐴′𝐵′𝐶′. Describe a sequence of rigid motions that takes 𝐴 to 𝐴′, 𝐵 to 𝐵′, and 𝐶 to 𝐶′.

5. Identify any angles of rotation that create symmetry.

146 | Unit 2

Unit 2, Lesson 13: Solving Problems Using Angle Pair Relationships

Warm-Up: Math Talk: Angle Relationships Lines ℓ and 𝑚 are parallel. Mentally evaluate the measure 𝑥 in each figure.

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Figure D

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Figure C

Figure B

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Figure A

Unit 2 | 147

Collaborative Activity: Solving Problems Using Angle Relationships Work with your partner to solve each problem. 1. A steel roller coaster in Sandusky, Ohio, uses support beams at various pressure points of the ride. Some of the support beams are shown in the diagram as 𝐾𝐿, 𝐿𝑅, and 𝐹𝑅.

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Determine the value of 𝑥 that shows that 𝐾𝐿 ∥ 𝐹𝑅, if 𝑚∠𝐾𝐿𝑅 = (7𝑥 − 5.6)° and 𝑚∠𝐹𝑅𝐿 = (3𝑥 + 5.6)°.

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2. Most of the streets in Miami are on a grid. Beacom Boulevard is one of the few streets that does not run north to south or east to west. The diagram shows a map of Beacom Boulevard, 𝐻𝐺, between SW 4th Street, 𝑊𝑇, and SW 5th Street, 𝐵𝑅.

a. Given that 𝑊𝑇 ∥ 𝐵𝑅, determine the value of 𝑥 if 𝑚∠𝐻𝑀𝐵 = (2𝑥 − 7)°, 𝑚∠𝑊𝐾𝐺 = (4𝑥 + 25)°, and 𝑚∠𝐾𝑀𝐵 = (6𝑥 − 29)°.

148 | Unit 2

b. Determine the measure of each angle. 𝑚∠𝐻𝑀𝐵 𝑚∠𝐻𝑀𝑅

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𝑚∠𝑇𝐾𝑀

3. Lines 𝐾𝑅, 𝑃𝑌, and 𝑀𝑇 are shown with 𝐿𝑊 where 𝑚∠𝐿𝑊𝐻 = �4𝑦 − 9 1 �°, 3

𝑚∠𝑊𝐿𝐻 = (3𝑦 + 3)°, and 𝑚∠𝐿𝐻𝑊 = 𝑚∠𝑊𝐿𝐻.

a. Determine the value of 𝑦 that shows that 𝑃𝑌 ∥ 𝐾𝑅.

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b. Determine the measure of each angle. 𝑚∠𝑅𝐿𝑊

𝑚∠𝑃𝐻𝑇

𝑚∠𝐾𝐿𝑀

Unit 2 | 149

4. Sheridan Street and Richmond Street are parallel, and both are intersected by University Boulevard.

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a. Determine the value of 𝑥.

5. Lines 𝑋𝐵, 𝑇𝑉, and 𝑅𝑊 are shown. Lines 𝑋𝐵 and 𝑇𝑉 are parallel, 𝑅𝑊 is a transversal,𝑚∠𝑋𝐻𝑅 = (9𝑥 + 21)°, and 𝑚∠𝐻𝐾𝑉 = (4𝑥 + 3)°.

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Two cars are heading west, one on Sheridan Street and one on Richmond Street. Both cars turn north onto University Boulevard. What is the measure of the angle of their turn onto University Boulevard?

b. Find the measure of each angle.

𝑚∠𝑋𝐻𝑅

𝑚∠𝑇𝐾𝑊

150 | Unit 2

𝑚∠𝐻𝐾𝑉

𝑚∠𝐵𝐻𝐾

Lesson Summary Angle Relationship

Angle Measures

Vertical angles

Congruent

Corresponding angles

Congruent

Alternate interior angles

Congruent

Alternate exterior angles

Congruent

Consecutive interior angles

Supplementary

Consecutive exterior angles

Supplementary

Practice Problems

a. Find the value of 𝑥.

1. Line 𝑆𝑌 is parallel to 𝐻𝑍 , 𝑚∠𝑌𝑅𝐶 = 64°, and 𝑚∠𝑅𝐶𝐻 = (8𝑥 − 4)°.

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Angle pairs created when parallel lines are intersected by a transversal are either congruent, supplementary, or neither. These relationships can be used to set up equations to solve problems in mathematical and real-world contexts. The table summarizes the relationships between angle pairs.

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b. Find the measure of each angle. 𝑚∠𝐹𝑅𝑆 =

𝑚∠𝑌𝑅𝐹 =

𝑚∠𝐻𝐶𝐴 =

𝑚∠𝑍𝐶𝐴 =

Unit 2 | 151

2. Harmon Lane, 𝐶𝐻 , crosses Camelot Drive, 𝑀𝑊, and University Drive, 𝑇𝐺, 𝑚∠𝑇𝑃𝐷 = 55°, and 𝑚∠𝑀𝐷𝑃 = (5𝑥 − 16)°.

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Determine the value of 𝑥 that shows 𝑀𝑊 ∥ 𝑇𝐺.

Review Problems

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3. Lines 𝐴𝐵 and 𝐵𝐶 are perpendicular. The dashed rays bisect angles 𝐴𝐵𝐷 and 𝐶𝐵𝐷. Explain why the measure of angle 𝐸𝐵𝐹 is 45°.

4. Select all the angles of rotation that produce symmetry for this flower.

□ 45° □ 60° □ 90° □ 120° □ 135° □ 150° □ 180°

152 | Unit 2

Unit 2, Lesson 14: Angle Relationship Proofs – Part 1

Warm-Up: Angle Pair Relationships

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1. The image shows 2 streets intersected by a third street. Various points of interest are located at the intersections.

In the table below, match the listed points of interest with their angle relationships.

Points of Interest

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Museum and Worship Center

Angle Relationship

A. Alternate interior angles

Bank and Day Care

B. Consecutive interior angles

Park and Coffee Shop

C. Corresponding angles

Fire Station and Worship Center

D. Vertical angles

Community Center and Day Care

E. Alternate exterior angles

Bank and Museum

F. Linear pair

Unit 2 | 153

Exploration Activity: Proofs with Parallel Lines and Transversals 1. Contradictory information is given for the figure. Line 𝐾𝐸 is parallel to 𝐻𝐷, ∠𝐾𝑅𝐴 ≅ ∠𝐵𝐺𝐸, ∠𝑇𝑅𝐺 is a supplement of ∠𝑅𝐺𝑊, and 𝑚∠𝐷𝑊𝑍 = 87°.

b. How do you know it’s contradictory?

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a. Which piece of information is contradictory?

c. With a partner, share the contradiction you each identified. Discuss any differences.

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d. Create four different pieces of information about this figure using 𝐴𝑉 ∥ 𝐵𝑍. One of the pieces of information should be contradictory. 1.

e. Trade information with a partner, and see if you can find the contradiction in each other’s information. 154 | Unit 2

2. Terrence was organizing a card sort for the proof, but his cards fell on the floor and are now out of order. Given: 𝑃𝑇 ∥ 𝑅𝐻 and ∠𝐶𝐵𝐺 ≅ ∠𝐶𝑌𝐺 Statements

A. 𝑚∠𝐶𝐵𝐺 = 𝑚∠𝐶𝑌𝐺

C. ∠𝐶𝐵𝐺 ≅ ∠𝐶𝑌𝐺

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Prove: 𝑊𝑍 ∥ 𝐴𝑁

B. ∠𝐶𝐵𝐺 and ∠𝐵𝐺𝑌 are supplementary. D. ∠𝐶𝑌𝐺 and ∠𝐵𝐺𝑌 are supplementary. E. 𝑊𝑍 ∥ 𝐴𝑁

G. 𝑃𝑇 ∥ 𝑅𝐻

F. 𝑚∠𝐶𝐵𝐺 + 𝑚∠𝐵𝐺𝑌 = 180°

H. 𝑚∠𝐶𝑌𝐺 + 𝑚∠𝐵𝐺𝑌 = 180°

Reasons

T. Given

N. Definition of congruence L. Substitution property of equality

P. Given

S. Definition of supplementary angles

R. If 2 parallel lines are intersected by a transversal, then the consecutive interior angles are supplementary.

M. If 2 lines are intersected by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

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K. Definition of supplementary angles

Work with your partner to place the statements and reasons (identified by letter) in the correct order.

Statement

Reason

Unit 2 | 155

3. Work with your partner to complete the proof in a different way. Given: 𝑃𝑇 ∥ 𝑅𝐻 and ∠𝐶𝐵𝐺 ≅ ∠𝐶𝑌𝐺

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Prove: 𝑊𝑍 ∥ 𝐴𝑁

Statement

Reason

1. Given

1. 𝑃𝑇 ∥ 𝑅𝐻

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3. ∠𝐶𝐵𝐺 ≅ ∠𝐶𝑌𝐺

2. If 2 parallel lines are intersected by a transversal, then corresponding angles are congruent.

4. Transitive property of congruence

5. 𝑊𝑍 ∥ 𝐴𝑁

3. Given

2. ∠𝐶𝐵𝐺 ≅ ∠

156 | Unit 2

Guided Activity: Proofs Involving Angle Relationships 1. Complete the flowchart proof. Given: ∠𝑅𝑁𝐷 and ∠𝐹𝐷𝑃 are supplementary.

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Prove: 𝐺𝑅 ∥ 𝐹𝐾

Unit 2 | 157

2. Complete the paragraph proof by choosing from the bank of statements and reasons below. Given: 𝑃𝑇 ∥ 𝑅𝐻 and ∠𝑃𝐶𝑊 ≅ ∠𝐻𝐺𝑁

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Prove: 𝑊𝑍 ∥ 𝐴𝑁

It is given that

. Therefore,

∠𝑅𝐵𝐶 ≅ ∠𝑃𝐶𝑊, because if 2 parallel lines are intersected by a transversal,

It is also given that

then

. By the

, it can be written that ∠𝑅𝐵𝐶 ≅ ∠𝐻𝐺𝑁.

Therefore, it can be concluded that 𝑊𝑍 ∥ 𝐴𝑁 , because if 2 lines are intersected by a

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transversal so that

, then the lines are

parallel.

Statements and Reasons

• 𝑃𝑇 ∥ 𝑅𝐻

• Definition of congruence

• ∠𝑃𝐶𝑊 ≅ ∠𝐻𝐺𝑁

• Definition of supplementary angles

• ∠𝐺𝑁𝐷

• Alternate exterior angles are congruent.

• Transitive property of congruence

• ∠𝑍𝐵𝐺 ≅ ∠𝐻𝐺𝑁

• Corresponding angles are congruent.

• ∠𝐹𝐷𝑃

• Alternate interior angles are congruent.

• 𝑊𝑍 ∥ 𝐴𝑁

158 | Unit 2

Lesson Summary A mathematical proof is an argument that shows something is true. A proof should begin with an assumption, assertion, or given statement. An assertion is a statement that you think is true but have not yet proved.

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The hypothesis is the given, or what is known to be true. The conclusion will need to be shown, or proven. Some common ways to write proofs include paragraph proofs, twocolumn proofs, and flowchart proofs. • A paragraph proof is a narrative that gives the steps with justifications that prove a statement is true. • A two-column proof consists of two columns: a chronological list of steps, which are called statements, and their justifications, which are called reasons.

• A flowchart proof shows the structure of the proof using boxes and connecting arrows. The justifications are written beside or below each box.

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When writing proofs, it’s important to remember that in geometry, there is a formal difference between equality and congruence. Equality refers to measurements and is expressed in statements such as 𝑚∠𝑀𝐻𝑅 = 𝑚∠𝐶𝑇𝑃. On the other hand, congruence means the shape is the same, which can be shown using rigid motions. To express congruence between the angles in the previous equality statement, the congruence statement ∠𝑀𝐻𝑅 ≅ ∠𝐶𝑇𝑃 is used.

• If 2 figures have equal measurements, then by the definition of congruence, the figures are congruent. Often, these 2 statements would follow each other in a proof.

• If congruence is shown through a series of transformations, then the definition of congruence in terms of rigid motions is applied.

Unit 2 | 159

Practice Problems 1. Use the word bank to complete the paragraph proof. Given: 𝑆𝑀 ∥ 𝑊𝑃

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Prove: ∠𝑆𝑀𝐾 and ∠𝑀𝑃𝐷 are supplementary.

It is given that 𝑆𝑀 ∥ 𝑊𝑃. Therefore, ∠𝑆𝑀𝐾 and ∠𝑊𝑃𝐹 are supplementary because

if 2 parallel lines are intersected by a transversal, then

angles are supplementary. By definition of

𝑚∠𝑆𝑀𝐾 + 𝑚∠𝑊𝑃𝐹 = 180°. ∠𝑊𝑃𝐹 ≅ ∠𝑀𝑃𝐷 because

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are congruent; therefore, 𝑚∠𝑊𝑃𝐹 = 𝑚∠𝑀𝑃𝐷 by definition of congruence. By the

, 𝑚∠𝑆𝑀𝐾 + 𝑚∠𝑀𝑃𝐷 = 180°. Therefore,

∠𝑆𝑀𝐾 and ∠𝑀𝑃𝐷 are supplementary by definition of supplementary angles. Word Bank

vertical angles

alternate exterior angles

160 | Unit 2

congruent

substitution property of equality

supplementary angles

transitive property of equality

2. Complete the two-column proof. Given: ∠𝐺𝐾𝑀 ≅ ∠𝐹𝐷𝐾

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Prove: ∠𝐻𝐾𝐷 ≅ 𝐹𝐷𝐾

Statement

Reason

1. Given

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Review Problems

1. ∠𝐺𝐾𝑀 ≅ ∠𝐹𝐷𝐾

3. In the figure shown, lines 𝑓 and 𝑔 are parallel. Select the angle that is congruent to angle 1. A. Angle 2

B. Angle 6 C. Angle 7 D. Angle 8

Unit 2 | 161

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4. Angle 𝐵𝐷𝐸 is congruent to angle 𝐵𝐴𝐶. Name another pair of congruent angles. Explain how you know.

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5. Quadrilateral 𝐴𝐵𝐶𝐷 is congruent to quadrilateral 𝐴′𝐵′𝐶′𝐷′. Describe a sequence of rigid motions that takes 𝐴 to 𝐴′, 𝐵 to 𝐵′, 𝐶 to 𝐶′, and 𝐷 to 𝐷′.

162 | Unit 2

Unit 2, Lesson 15: Angle Relationship Proofs – Part 2

Warm-Up: Would You Rather 1. The meals at 2 restaurants range from $8 to $25. Servers are paid as follows. Restaurant B

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Restaurant A

$18 per hour No tipping allowed.

$10.50 per hour Tipping encouraged.

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Would you rather work as a server at restaurant A or at restaurant B? Justify your choice.

Unit 2 | 163

Collaborative Activity: Card Sort 1. With your group, sort through the statements and reasons for each proof. Then, write each proof in the tables provided.

Given: 𝑊𝑍 ∥ 𝐴𝑁 and ∠𝑊𝐶𝑌 ≅ ∠𝑌𝐺𝐻

Reason

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Statement

Prove: 𝑃𝑇 ∥ 𝑅𝐻

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Proof A

164 | Unit 2

Proof B Given: ∠𝐺𝑁𝐷 and ∠𝑃𝐷𝐾 are supplementary.

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Prove: 𝐺𝑅 ∥ 𝐹𝐾

Reason

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Statement

Unit 2 | 165

Proof C Given: 𝐵𝑃 ⊥ 𝐺𝑅 and 𝐺𝑅 ∥ 𝐹𝐾

B G

Prove: 𝐵𝑃 ⊥ 𝐹𝐾

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Reason

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Statement

166 | Unit 2

Proof D Given: 𝐺𝑅 ∥ 𝐹𝐾

B N

Prove: ∠𝐺𝑁𝐷 and ∠𝑃𝐷𝐾 are supplementary.

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Reason

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Statement

Unit 2 | 167

Proof E Given: 𝐺𝑅 ⊥ 𝐵𝑃 and 𝐹𝐾 ⊥ 𝐵𝑃

Prove: 𝐺𝑅 ∥ 𝐹𝐾

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Reason

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Statement

168 | Unit 2

Proof F Given: ∠𝑇𝐶𝑆 ≅ ∠𝐻𝑆𝐶 and ∠𝐻𝑆𝐾 ≅ ∠𝑆𝐾𝐺 Prove: 𝑅𝑇 ∥ 𝑍𝐺 Reason

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Statement

Lesson Summary

In many situations, it’s important to understand the reasons why an idea is true. Proofs can be used to convince others that a statement is true. When writing proofs, it’s important to label diagrams and make precise arguments using geometric facts. So far in this course, this has included using transformations or angle relationships to justify relationships. Many true statements have multiple explanations or reasons. It’s important to pick the reason that will most logically bring the proof to its conclusion. Geometric proofs do not necessarily need to be in a specific order, as long as the reasoning leads to the goal of the statement you’re trying to prove. Proofs can be completed in different ways, but as long as the statements lead to a logical conclusion, the proof is valid. © Accelerate Learning Inc. - All Rights Reserved

Unit 2 | 169

Practice Problems 1. Complete the two-column proof. Given: ∠𝐻𝐾𝑀 and ∠𝐹𝐷𝐾 are supplementary.

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Prove: 𝐺𝐻 ∥ 𝐹𝐶

Statement

Reason

𝑚∠𝐻𝐾𝑀 + 𝑚∠𝐹𝐷𝐾 = 180°

∠𝐻𝐾𝑀 and ∠𝐵𝐷𝐶 are supplementary.

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∠𝐺𝐾𝐷 ≅ ∠𝐻𝐾𝑀

Vertical angles are congruent.

Definition of congruence

𝑚∠𝐺𝐾𝐷 + 𝑚∠𝐹𝐷𝐾 = 180° ∠𝐺𝐾𝐷 and ∠𝐹𝐷𝐾 are supplementary.

Definition of supplementary angles

𝐺𝐻 ∥ 𝐹𝐶 170 | Unit 2

2. Alejandra completed a two-column proof as shown. Given: 𝑀𝑃 ∥ 𝑇𝑊 and 𝐻𝑇 ∥ 𝐾𝑉

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Prove: ∠𝑇𝑉𝑍 ≅ ∠𝐻𝑁𝑅

Statement

If 2 parallel lines are intersected by a transversal, then alternate exterior angles are congruent.

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𝐻𝑇 ∥ 𝐾𝑉

∠𝑇𝑉𝑍 ≅ ∠𝐾𝑃𝑅

Given

𝑀𝑃 ∥ 𝑇𝑊

Reason

∠𝐻𝑁𝑅 ≅ ∠𝐾𝑃𝑅

Given

If 2 parallel lines are intersected by a transversal, then corresponding angles are congruent.

Transitive property of congruence

∠𝑇𝑉𝑍 ≅ ∠𝐻𝑁𝑅

Alejandra says this is the only way to prove ∠𝑇𝑉𝑍 ≅ ∠𝐻𝑁𝑅. Explain whether you agree with her.

Unit 2 | 171

Review Problems 3. In the figure shown, angle 3 is congruent to angle 6. Select all statements that must be true.

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□ Lines 𝑓 and 𝑔 are parallel. □ Angle 2 is congruent to angle 6. □ Angle 2 and angle 5 are supplementary. □ Angle 1 is congruent to angle 7. □ Angle 4 is congruent to angle 6.

4. Lines 𝐴𝐷 and 𝐸𝐶 meet at point 𝐵.

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Give an example of a rotation using an angle greater than 0° and less than 360° that takes both lines to themselves. Explain why your rotation works.

172 | Unit 2

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Unit 3: Triangle Congruence

Unit 3 | 173

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Unit 3, Lesson 1: Congruent Parts

Warm-Up: Notice and Wonder: Transformed Rectangles

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What do you notice? What do you wonder?

Unit 3 | 175

Exploration Activity: If We Know This, Then We Know That

∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹

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Triangle 𝐴𝐵𝐶 is congruent to ∆𝐷𝐸𝐹.

1. Find a sequence of rigid motions that takes ∆𝐴𝐵𝐶 to ∆𝐷𝐸𝐹.

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2. What is the image of 𝐵𝐶 after that transformation?

3. Explain how you know those segments are congruent.

4. Justify that angle ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹.

176 | Unit 3

Exploration Activity: Which Triangles Are Congruent?

This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

1. Triangle 𝑃𝑄𝑅 is congruent to which triangle? Explain your reasoning.

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Here are three triangles.

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2. Show a sequence of rigid transformations that takes ∆𝑃𝑄𝑅 to that triangle. Draw each step of the transformation.

3. Explain why there can’t be a rigid transformation to the other triangle.

Unit 3 | 177

Lesson Summary For a rigid transformation that takes 1 figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. Corresponding parts could be angles, points, or sides. • If 2 figures are not congruent, then there is not a rigid transformation that takes 1 figure onto the other.

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• If 2 figures are congruent, then there is a rigid transformation that takes 1 figure onto the other. The same rigid transformation can be applied to individual parts of the figure, such as segments and angles, because rigid transformations move every point on the plane. Therefore, the corresponding parts of 2 congruent figures are congruent to each other. Corresponding parts are used when trying to prove whether 2 figures are congruent. For this reason, it’s important to name congruent figures so that it’s clear from the names which parts correspond, and this makes it easier to check whether the figures are congruent.

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For example, in the image shown, 𝐴𝐵 appears to be congruent to 𝐷𝐸 and 𝐸𝐹 appears to be congruent to 𝐵𝐶. Therefore, it makes more sense to conjecture that ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 than to conjecture ∆𝐴𝐵𝐶 ≅ ∆𝐹𝐷𝐸. Making such conjectures will be important for proving whether figures are, in fact, congruent.

Practice Problems

1. Triangle 𝐻𝐸𝐹 is the image of ∆𝐹𝐺𝐻 after a 180° rotation about point 𝐾. Select all statements that must be true.

□ ∆𝐹𝐺𝐻 ≅ ∆𝐹𝐸𝐻 □ ∆𝐸𝐹𝐻 ≅ ∆𝐺𝐹𝐻 □ ∠𝐾𝐻𝐸 ≅ ∠𝐾𝐹𝐺 □ ∠𝐺𝐻𝐾 ≅ ∠𝐾𝐻𝐸 □ 𝐸𝐻 ≅ 𝐹𝐺 □ 𝐺𝐻 ≅ 𝐸𝐹

178 | Unit 3

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2. Reflect right triangle 𝐴𝐵𝐶 across 𝐵𝐶. Classify ∆𝐴𝐶𝐴′ according to its side lengths. Explain how you know.

Review Problems

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3. Elena needs to prove ∠𝐵𝐸𝐷 and ∠𝐵𝐶𝐴 are congruent. Provide reasons to support each of her statements.

a. Line 𝑚 is parallel to line 𝑙.

b. Angles 𝐵𝐸𝐷 and 𝐵𝐶𝐴 are congruent.

Unit 3 | 179

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4. Triangles 𝐹𝐴𝐷 and 𝐷𝐶𝐸 are translations of ∆𝐴𝐵𝐶.

Select all the statements that must be true.

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□ Points 𝐵, 𝐴, and 𝐹 are collinear. □ The measure of ∠𝐵𝐶𝐴 is the same as the measure of ∠𝐶𝐸𝐷. □ Line 𝐴𝐷 is parallel to 𝐵𝐶. □ The measure of ∠𝐶𝐸𝐷 is the same as the measure of ∠𝐹𝐴𝐷. □ The measure of ∠𝐷𝐴𝐶 is the same as the measure of ∠𝐵𝐶𝐴. □ Triangle 𝐴𝐷𝐶 is a reflection of ∆𝐹𝐴𝐷.

180 | Unit 3

Unit 3, Lesson 2: Congruent Triangles – Part 1

Warm-Up: True or . . . Sometimes True?

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This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

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1. What must be true?

If ∆𝐴𝐵𝐶 is congruent to ∆𝐴′𝐵′𝐶′. . .

2. What could possibly be true?

3. What definitely can’t be true?

Unit 3 | 181

Collaborative Activity: Invisible Triangles Player 1: You are the transformer. Take the transformer card. Player 2: Select a triangle card. Do not show it to anyone. Study the diagram to figure out which sides and which angles correspond. Tell Player 1 what you have figured out.

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Player 1: Take notes about what they tell you so that you know which parts of their triangles correspond. Think of a sequence of rigid motions you could tell your partner to get them to take one of their triangles onto the other. Be specific in your language. The notes on your card can help with this.

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Player 2: Listen to the instructions from the transformer. Use tracing paper to follow their instructions. Draw the image after each step. Let them know when they have lined up 1, 2, or all 3 vertices on your triangles.

182 | Unit 3

Collaborative Activity: Why Do They Coincide? Noah and Priya were playing Invisible Triangles. For card 3, Priya told Noah that in ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹: • 𝐴𝐵 ≅ 𝐷𝐸

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• 𝐴𝐶 ≅ 𝐷𝐹 • 𝐵𝐶 ≅ 𝐸𝐹

• ∠𝐴 ≅ ∠𝐷 • ∠𝐵 ≅ ∠𝐸 • ∠𝐶 ≅ ∠𝐹

Here are the steps Noah had to tell Priya to do before all 3 vertices coincided.

• Translate ∆𝐴𝐵𝐶 by the directed line segment from 𝐴 to 𝐷.

• Rotate the image, ∆𝐴′𝐵′𝐶′, using 𝐷 as the center, so that 𝐴″𝐵″ and 𝐷𝐸 line up.

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• Reflect the image, ∆𝐴″𝐵″𝐶″, across 𝐷𝐸.

After those steps, the triangles were lined up perfectly. Now Noah and Priya are working on explaining why their steps worked, and they need some help. Answer their questions.

First, we translate ∆𝐴𝐵𝐶 by the directed line segment from 𝐴 to 𝐷. Point 𝐴′ will coincide with 𝐷 because we defined our transformation that way. Then, rotate the image, ∆𝐴′𝐵′𝐶′, by the ∠𝐵′𝐷𝐸, so that 𝐴″𝐵″ and 𝐷𝐸 line up. 1. We know that 𝐴″𝐵″ and 𝐷𝐸 line up because we said they had to, but why do points 𝐵″ and 𝐸 have to be in the exact same place? © Accelerate Learning Inc. - All Rights Reserved

Unit 3 | 183

2. Finally, reflect the image, ∆𝐴″𝐵″𝐶″ across 𝐷𝐸.

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a. How do we know that now, the image of ray 𝐴″𝐶″ and ray 𝐷𝐹 will line up?

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b. How do we know that the image of point 𝐶″ and point 𝐹 will line up exactly?

184 | Unit 3

Lesson Summary If all corresponding parts of 2 triangles are congruent, then 1 triangle can be taken exactly onto the other triangle using a sequence of translations, rotations, and reflections. The congruence of corresponding parts justifies that the vertices of the triangles will line up exactly.

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One of the most common ways to line up the vertices is through a translation to get 1 pair of vertices to coincide, followed by a rotation to get a second pair of vertices to coincide, and if needed, a reflection to get the third pair of vertices to coincide. There are multiple ways to justify why the vertices must line up if the triangles are congruent, but 1 method of doing so is described and shown.

• First, translate ∆𝐴𝐵𝐶 by the directed line segment from 𝐴 to 𝐷. Points 𝐴 and 𝐷 coincide after translating because the translation was defined that way. Then, rotate the image of ∆𝐴𝐵𝐶 using point 𝐷 as the center, so that 𝐴′𝐵′ and 𝐷𝐸 line up.

𝐴𝐵 ≅ 𝐷𝐸, 𝐵𝐶 ≅ 𝐸𝐹, 𝐴𝐶 ≅ 𝐷𝐹, and ∠𝐴 ≅ ∠𝐷, ∠𝐵 ≅ ∠𝐸, ∠𝐶 ≅ ∠𝐹

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• Line segments 𝐴′𝐵′ and 𝐷𝐸 line up because that is how the rotation is defined. The distance 𝐴𝐵 is the same as the distance 𝐷𝐸 because translations and rotations don’t change distances. Since points 𝐵′ and 𝐸 are the same distance along the same ray from 𝐷, they have to be in the same place.

• If necessary, reflect ∆𝐴′𝐵′𝐶′ across 𝐷𝐸 so that the image of 𝐶 is on the same side of 𝐷𝐸 as 𝐹. Angle 𝐴 is congruent to ∠𝐷 because translation, rotation, and reflection don’t change angle measures.

• Point 𝐶″ must be on 𝐷𝐹 since both 𝐶″ and 𝐹 are on the same side of 𝐷𝐸 and make the same angle with it at point 𝐷. The distance 𝐴𝐶 is the same as the distance 𝐷𝐹, so that means 𝐶″ is the same distance from 𝐴″ as 𝐹 is from 𝐷 because translations and rotations preserve distance. Since 𝐹 and 𝐶″ are the same distance along the same ray from 𝐷, they have to be in the same place. © Accelerate Learning Inc. - All Rights Reserved

Unit 3 | 185

Practice Problems

□ Angle 𝐴 coincides with ∠𝐹. □ Angle 𝐵 coincides with ∠𝐷. □ Segment 𝐴𝐶 coincides with 𝐸𝐹. □ Segment 𝐵𝐶 coincides with 𝐸𝐷. □ Segment 𝐴𝐵 coincides with 𝐸𝐷.

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1. Triangle 𝐴𝐵𝐶 is congruent to ∆𝐸𝐷𝐹, so Kiran knows that there is a sequence of rigid motions that takes ∆𝐴𝐵𝐶 to ∆𝐸𝐷𝐹. Select all true statements after the transformations.

2. A rotation by ∠𝐴𝐶𝐸 using point 𝐶 as the center takes ∆𝐶𝐵𝐴 onto ∆𝐶𝐷𝐸.

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a. Explain why the image of 𝐶𝐴 lines up with 𝐶𝐸 .

b. Explain why the image of 𝐴 coincides with 𝐸.

c. Is ∆𝐶𝐵𝐴 ≅ ∆𝐶𝐷𝐸? Explain your reasoning.

186 | Unit 3

3. The triangles are congruent. Which sequence of rigid motions will take ∆𝑋𝑌𝑍 onto ∆𝐵𝐶𝐴?

A. Translate ∆𝑋𝑌𝑍 using directed line segment 𝑌𝐶. Rotate ∆𝑋′𝑌′𝑍′ using 𝐶 as the center so that 𝑋′ coincides with 𝐵. Reflect ∆𝑋″𝑌″𝑍″ across 𝐶𝐵.

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B. Translate ∆𝑋𝑌𝑍 using directed line segment 𝑌𝐶. Rotate ∆𝑋′𝑌′𝑍′ using 𝐶 as the center so that 𝑋′ coincides with 𝐵. Reflect ∆𝑋″𝑌″𝑍″ across 𝐴𝐶. C. Translate ∆𝑋𝑌𝑍 using directed line segment 𝑌𝐶. Rotate ∆𝑋′𝑌′𝑍′ using 𝐶 as the center so that 𝑋′ coincides with 𝐴. Reflect ∆𝑋″𝑌″𝑍″ across 𝐶𝐵.

Review Problems

D. Translate ∆𝑋𝑌𝑍 using directed line segment 𝑌𝐶. Rotate ∆𝑋′𝑌′𝑍′ using 𝐶 as the center so that 𝑋′ coincides with 𝐴. Reflect ∆𝑋″𝑌″𝑍″ across 𝐴𝐶.

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4. Triangle 𝐻𝐸𝐹 is the image of ∆𝐹𝐺𝐻 after a 180° rotation around point 𝐾. Select all statements that must be true.

□ ∆𝐻𝐺𝐹 ≅ ∆𝐹𝐸𝐻 □ ∆𝐺𝐹𝐻 ≅ ∆𝐸𝐹𝐻 □ ∠𝐾𝐻𝐸 ≅ ∠𝐾𝐻𝐺 □ ∠𝐺𝐻𝐾 ≅ ∠𝐸𝐹𝐾 □ 𝐸𝐻 ≅ 𝐺𝐻 □ 𝐻𝐺 ≅ 𝐹𝐸 □ 𝐹𝐻 ≅ 𝐻𝐹

Unit 3 | 187

□ Segment 𝐴𝐶 is congruent to 𝐸𝐹. □ Segment 𝐵𝐶 is congruent to 𝐸𝐹. □ Angle 𝐵𝐴𝐶 is congruent to ∠𝐸𝐷𝐹. □ Angle 𝐵𝐶𝐴 is congruent to ∠𝐸𝐷𝐹. □ Angle 𝐶𝐵𝐴 is congruent to ∠𝐹𝐸𝐷.

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5. Triangle 𝐴𝐵𝐶 is congruent to ∆𝐷𝐸𝐹. Select all the statements that are a result of corresponding parts of congruent triangles being congruent.

6. When ∆𝐴𝐵𝐶 is reflected across 𝐴𝐵, the image is ∆𝐴𝐵𝐷. Why is ∠𝐴𝐶𝐷 ≅ ∠𝐴𝐷𝐵?

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A. Corresponding parts of congruent figures are congruent. B. Congruent parts of congruent figures are corresponding. C. Segment 𝐴𝐵 is a perpendicular bisector of 𝐷𝐶.

D. An isosceles triangle has a pair of congruent angles.

188 | Unit 3

Unit 3, Lesson 3: Congruent Triangles – Part 2

Warm-Up: Make That Triangle

• Angle 𝐴 is 40°.

• Angle 𝐵 is 20°.

• Angle 𝐶 is 120°.

• Segment 𝐴𝐵 is 5 centimeters (cm). • Segment 𝐵𝐶 is 3.7 cm.

• Segment 𝐴𝐶 is 2 cm.

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Draw triangle 𝐴𝐵𝐶 with these measurements:

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Highlight each piece of given information that you used. Check your triangle to make sure the remaining measurements match.

Unit 3 | 189

Exploration Activity: Info Gap: Too Much Information Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the data card:

1. Silently read your card and think about what information you need to answer the question.

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1. Silently read the information on your card.

If your teacher gives you the problem card:

2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. 2. Ask your partner for the specific Only give information that is on your information that you need. card. (Do not figure out anything for your partner!) 3. Explain to your partner how you are using the information to solve the 3. Before telling your partner the problem. information, ask “Why do you need to know (that piece of information)?” 4. When you have enough information, share the problem card with your 4. Read the problem card, and solve the partner, and solve the problem problem independently. independently.

5. Read the data card, and discuss your reasoning.

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5. Share the data card, and discuss your reasoning.

190 | Unit 3

Exploration Activity: Too Little Information?

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Jada and Tyler were playing the Info Gap, using Card 3.

Tyler asked, “Can I have 2 sides and an angle?”

Jada told Tyler that one angle was 16°, one side was 5 cm, and one side was 4 cm. Here is the triangle Tyler made:

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1. Is Tyler’s triangle congruent to the triangle on the Data Card?

2. Did Tyler do anything that didn’t match Jada’s instructions?

3. How could Tyler have made a more specific request for 2 sides and an angle so that his triangle was guaranteed to match Jada’s?

Unit 3 | 191

Lesson Summary If every pair of corresponding parts of 2 triangles is congruent, then the 2 triangles are congruent. However, it is not necessary to have information about every pair of corresponding parts to determine if triangles are congruent. For example, if 2 angles of a triangle are 30° and 60°, it can be concluded that the third angle is 90° because the interior angles of a triangle sum to 180°.

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Figuring out which sets of measurements are enough to draw a complete triangle relates to which sets of measurements are sufficient to prove triangles are congruent. For example, if 2 side lengths of a triangle are known along with the measure of the angle formed where they meet, there is only 1 possible line segment length that could join the other endpoints of the 2 original sides.

Shown are 3 sets of measurements that appear to be enough information to prove that the 2 triangles are congruent.

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• Two pairs of corresponding sides are congruent, and the angles between those sides are congruent.

• Two pairs of corresponding angles are congruent, and the sides between those angles are congruent.

• Three pairs of corresponding sides are congruent.

192 | Unit 3

Practice Problems 1. Match each statement using only the information shown in the pairs of congruent triangles. 1.

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A. In the 2 triangles there are 3 pairs of congruent sides.

C. The 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle.

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B. The 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle.

2. Sketch the unique triangles that can be made with angle measures 40° and 100° and side length 3. How do you know you have sketched all possibilities?

Unit 3 | 193

Review Problems

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3. What is the least amount of information that you need to create a triangle congruent to this one?

4. Triangle 𝐴𝐵𝐶 is congruent to triangle 𝐸𝐷𝐹. So, Mai knows that there is a sequence of rigid motions that takes 𝐴𝐵𝐶 to 𝐸𝐷𝐹. Select all true statements after the transformations.

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□ Angle 𝐴 coincides with angle 𝐸. □ Angle 𝐵 coincides with angle 𝐹. □ Segment 𝐴𝐵 coincides with segment 𝐸𝐹. □ Segment 𝐵𝐶 coincides with segment 𝐷𝐹. □ Segment 𝐴𝐶 coincides with segment 𝐸𝐷.

5. A rotation by ∠𝐴𝐶𝐸 using point 𝐶 as the center takes ∆𝐶𝐵𝐴 onto triangle ∆𝐶𝐷𝐸.

a. Explain why the image of segment 𝐶𝐵 lines up with segment 𝐶𝐷.

194 | Unit 3

b. Explain why the image of 𝐵 coincides with 𝐷.

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c. Is triangle 𝐴𝐵𝐶 congruent to triangle 𝐸𝐷𝐶? Explain your reasoning.

6. Triangle 𝐻𝐸𝐹 is the image of triangle 𝐻𝐺𝐹 after a reflection across line 𝐹𝐻. Select all statements that must be true.

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□ Triangle 𝐹𝐺𝐻 is congruent to triangle 𝐹𝐸𝐻. □ Triangle 𝐸𝐹𝐻 is congruent to triangle 𝐺𝐹𝐻. □ Angle 𝐻𝐹𝐸 is congruent to angle 𝐹𝐻𝐺. □ Angle 𝐸𝐹𝐺 is congruent to angle 𝐸𝐻𝐺. □ Segment 𝐸𝐻 is congruent to segment 𝐹𝐺. □ Segment 𝐺𝐻 is congruent to segment 𝐸𝐻.

7. This design began from the construction of a regular hexagon. Describe a rigid motion that will take the figure onto itself.

Unit 3 | 195

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Unit 3, Lesson 4: Side-Angle-Side Congruence

Warm-Up: Information Overload? Highlight each piece of given information that is used in the proof and each line in the proof where that piece of information is used.

• 𝐴𝐵 ≅ 𝐷𝐸

• ∠𝐴 ≅ ∠𝐷

• 𝐵𝐶 ≅ 𝐸𝐹

• ∠𝐶 ≅ ∠𝐹

• 𝐴𝐶 ≅ 𝐷𝐹

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Given:

• ∠𝐵 ≅ ∠𝐸

Proof:

1. Segments 𝐴𝐵 and 𝐷𝐸 are the same length so they are congruent. Therefore, there is a rigid motion that takes 𝐴𝐵 to 𝐷𝐸.

2. Apply that rigid motion to triangle 𝐴𝐵𝐶. The image of 𝐴 will coincide with 𝐷, and the image of 𝐵 will coincide with 𝐸.

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3. We cannot be sure that the image of 𝐶 coincides with 𝐹 yet. If necessary, reflect the image of ∆𝐴𝐵𝐶 across 𝐷𝐸 to be sure the image of 𝐶, which we will call 𝐶′, is on the same side of 𝐷𝐸 as 𝐹. (This reflection does not change the image of 𝐴 or 𝐵.)

4. We know the image of ∠𝐴 is congruent to ∠𝐷 because rigid motions don’t change the size of angles. 5. Point 𝐶′ must be on 𝐷𝐹 since both 𝐶′ and 𝐹 are on the same side of 𝐷𝐸, and make the same angle with it at 𝐷.

6. Segment 𝐷𝐶′ is the image of 𝐴𝐶 and rigid motions preserve distance, so they must have the same length. 7. We also know 𝐴𝐶 has the same length as 𝐷𝐹. So 𝐷𝐶′ and 𝐷𝐹 must be the same length.

8. Since 𝐶′ and 𝐹 are the same distance along the same ray from 𝐷, they have to be in the same place. 9. We have shown that a rigid motion takes 𝐴 to 𝐷, 𝐵 to 𝐸, and 𝐶 to 𝐹; therefore, ∆𝐴𝐵𝐶 is congruent to ∆𝐷𝐸𝐹. © Accelerate Learning Inc. - All Rights Reserved

Unit 3 | 197

Exploration Activity: Proving the Side-Angle-Side Triangle Congruence Theorem 1. Two triangles have 2 pairs of corresponding sides congruent, and the corresponding angles between those sides are congruent. Sketch 2 triangles that fit this description and label them 𝐿𝑀𝑁 and 𝑃𝑄𝑅, so that: • 𝐿𝑀 ≅ 𝑃𝑄

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• 𝐿𝑁 ≅ 𝑃𝑅

• ∠𝐿 ≅ ∠𝑃

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2. Use a sequence of rigid motions to take ∆𝐿𝑀𝑁 onto ∆𝑃𝑄𝑅. For each step, explain how you know that one or more vertices will line up.

3. Look back at the congruent triangle proofs from the Warm-Up. Do you have enough information here to use a proof that is like one you saw earlier? Use the proof to guide you in writing a proof for this situation.

198 | Unit 3

Collaborative Activity: What Can We Conclude about Isosceles Triangles? Mai and Kiran want to prove that in an isosceles triangle, the 2 base angles are congruent. Isosceles triangle 𝑃𝐴𝐵 is shown. 1. Draw the perpendicular bisector of 𝐵𝐴 through vertex 𝑃 of the isosceles triangle. Label the midpoint of 𝐵𝐴 as point 𝑄.

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2. Discuss with your partner what additional information can be concluded after drawing the perpendicular bisector.

3. Use options from the word bank to complete the proof. Word Bank ∠𝐴𝑃𝑄 ≅ ∠𝐵𝑃𝑄

𝑃𝐴 ≅ 𝑃𝐵

Since ∆𝑃𝐴𝐵 is an isosceles triangle,

definition of midpoint

side-angle-side congruence

reflexive property of congruence

∠𝑃𝐵𝑄 ≅ ∠𝑃𝐴𝑄

because of the definition of

isosceles triangle. Line segment 𝑃𝑄 bisects ∠𝐵𝑃𝐴 so that

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definition of an angle bisector. 𝑃𝑄 ≅ 𝑃𝑄 by the Therefore, ∆𝐴𝑃𝑄 ≅ ∆𝐵𝑃𝑄 by the

by the . .

4. Explain why the 2 base angles of an isosceles triangle are congruent.

Unit 3 | 199

Lesson Summary

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If all pairs of corresponding sides and angles in 2 triangles are congruent, then it is possible to find a rigid transformation that takes corresponding vertices onto one another. This proves that if 2 triangles have all pairs of corresponding sides and angles congruent, then the triangles must be congruent. But justifying that the vertices must line up does not require knowing all the pairs of corresponding sides and angles are congruent.

If all that is known is that 2 pairs of corresponding sides and the pair of corresponding angles between the sides are congruent, this is sufficient to prove the triangles are congruent. This is called the side-angle-side (SAS) triangle congruence theorem.

To determine if 2 triangles or 2 corresponding parts of triangles are congruent, first check whether the given information or the diagram indicates that 2 pairs of corresponding sides and the pair of corresponding angles between the sides are congruent. If that is the case, transformations that take 1 triangle onto the other triangle are not needed to prove congruence. Instead, the congruence of the 2 triangles can be justified by the SAS congruence theorem.

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Sometimes, however, to determine whether triangles are congruent, it may be necessary to add more lines to the diagram. Determine which properties those lines should have based on how they are drawn (An angle bisector? A perpendicular bisector? A line connecting 2 given points?). Mathematicians call these additional lines auxiliary lines because auxiliary means “providing additional help or support.”

An auxiliary line is an extra line drawn in a figure to reveal hidden structure.

These are lines that give extra help in seeing hidden structures within triangles.

200 | Unit 3

Practice Problems 1. Triangle 𝐷𝐴𝐶 is isosceles with congruent sides 𝐴𝐷 and 𝐴𝐶. Which additional given information is sufficient for showing that triangle 𝐷𝐵𝐶 is isosceles? Select all that apply.

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□ Line 𝐴𝐵 is an angle bisector of ∠𝐷𝐴𝐶. □ Angle 𝐵𝐴𝐷 is congruent to ∠𝐴𝐵𝐶. □ Angle 𝐵𝐷𝐶 is congruent to ∠𝐵𝐶𝐷. □ Angle 𝐴𝐵𝐷 is congruent to ∠𝐴𝐵𝐶. □ Triangle 𝐷𝐴𝐵 is congruent to ∆𝐶𝐴𝐵.

2. Tyler has written an incorrect proof to show that quadrilateral 𝐴𝐵𝐶𝐷 is a parallelogram. He knows segments 𝐴𝐵 and 𝐷𝐶 are congruent. He also knows ∠𝐴𝐵𝐶 and ∠𝐴𝐷𝐶 are congruent. Find the mistake in his proof.

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Segment 𝐴𝐶 is congruent to itself, so ∆𝐴𝐵𝐶 is congruent to ∆𝐴𝐷𝐶 by SAS triangle congruence theorem. Since the triangles are congruent, so are the corresponding parts, and so ∠𝐷𝐴𝐶 is congruent to ∠𝐴𝐶𝐵. In quadrilateral 𝐴𝐵𝐶𝐷, 𝐴𝐵 is congruent to 𝐶𝐷 and 𝐴𝐷 is parallel to 𝐶𝐵. Since 𝐴𝐷 is parallel to 𝐶𝐵, alternate interior ∠𝐷𝐴𝐶 and ∠𝐵𝐶𝐴 are congruent. Since alternate interior angles are congruent, 𝐴𝐵 must be parallel to 𝐶𝐷. Quadrilateral 𝐴𝐵𝐶𝐷 must be a parallelogram since both pairs of opposite sides are parallel.

Unit 3 | 201

3. Triangles 𝐴𝐶𝐷 and 𝐵𝐶𝐷 are isosceles. Angle 𝐵𝐴𝐶 has a measure of 18° and ∠𝐵𝐷𝐶 has a measure of 48°. Find the measure of ∠𝐴𝐵𝐷.

Review Problems

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𝐴𝐷 ≅ 𝐴𝐶 and 𝐵𝐷 ≅ 𝐵𝐶

4. Triangle 𝐴𝐵𝐶 is congruent to ∆𝐸𝐷𝐹. So, Priya knows that there is a sequence of rigid motions that takes ∆𝐴𝐵𝐶 to ∆𝐸𝐷𝐹. Select all true statements after the transformations.

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□ Segment 𝐴𝐵 coincides with 𝐸𝐹. □ Segment 𝐵𝐶 coincides with 𝐷𝐹. □ Segment 𝐴𝐶 coincides with 𝐸𝐷. □ Angle 𝐴 coincides with ∠𝐸. □ Angle 𝐶 coincides with ∠𝐹.

5. Triangle 𝐻𝐸𝐹 is the image of ∆𝐻𝐺𝐹 after a reflection across 𝐹𝐻. Write a congruence statement for the 2 congruent triangles.

202 | Unit 3

Unit 3, Lesson 5: Angle-Side-Angle Triangle Congruence

Warm-Up: Notice and Wonder: Assertion

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This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

What do you notice? What do you wonder?

Assertion: Through two distinct points passes a unique line. Two lines are said to be distinct if there is at least one point that belongs to one but not the other. Otherwise, we say the lines are the same. Lines that have no point in common are said to be parallel. Therefore, we can conclude: given two distinct lines, either they are parallel, or they have exactly one point in common. © Accelerate Learning Inc. - All Rights Reserved

Unit 3 | 203

Exploration Activity: Proving the Angle-Side-Angle Triangle Congruence Theorem

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1. Two triangles have 2 pairs of corresponding angles congruent, and the corresponding sides between those angles are congruent. Sketch 2 triangles that fit this description.

2. Label the triangles 𝑊𝑋𝑌 and 𝐷𝐸𝐹, so that angle 𝑊 is congruent to angle 𝐷, angle 𝑋 is congruent to angle 𝐸, and side 𝑊𝑋 is congruent to side 𝐷𝐸.

3. Use a sequence of rigid motions to take triangle 𝑊𝑋𝑌 onto triangle 𝐷𝐸𝐹. For each step, explain how you know that one or more vertices will line up.

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Collaborative Activity: Find the Missing Angle Measure

Lines ℓ and 𝑚 are parallel as shown, where ∠𝑎 = 42°.

1. Determine each angle measure. ∠𝑏 =

∠𝑓 = 204 | Unit 3

∠𝑐 =

∠𝑔 =

∠𝑑 = ∠ℎ =

∠𝑒 =

Guided Activity: What Can Be Concluded about Parallelograms? Quadrilateral 𝐴𝐵𝐶𝐷 is a parallelogram. By definition, that means 𝐴𝐵 ∥ 𝐷𝐶 and 𝐵𝐶 ∥ 𝐴𝐷.

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1. Sketch parallelogram 𝐴𝐵𝐶𝐷, and then draw an auxiliary line 𝐴𝐶 to show how 𝐴𝐵𝐶𝐷 can be decomposed into 2 triangles.

2. Complete the proof demonstrating that the 2 triangles you created are congruent. Given: 𝐴𝐵 ∥ 𝐷𝐶 and 𝐵𝐶 ∥ 𝐴𝐷

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴

𝐴𝐵 ∥ 𝐷𝐶

Statement

Reason Given

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∠𝐵𝐴𝐶 ≅ ∠𝐷𝐶𝐴 𝐵𝐶 ∥ 𝐴𝐷

Given

∠𝐴𝐶𝐵 ≅ ∠𝐶𝐴𝐷

Reflexive property of congruence

∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴

Unit 3 | 205

Lesson Summary If 2 triangles have 2 pairs of corresponding sides and the pair of corresponding angles between the sides are congruent, then the triangles must be congruent. But it might not always be given that 2 pairs of corresponding sides are congruent.

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For example, when proving that opposite sides of a parallelogram are congruent, there is only information about 1 pair of opposite sides. That is why there are different ways to prove congruent triangles other than side-angle-side (SAS) congruence explored in the previous lesson.

According to the angle-side-angle (ASA) triangle congruence theorem, if 2 pairs of corresponding angles and the pair of corresponding sides between the angles in 2 triangles are congruent, then the triangles must be congruent. This congruence relationship is demonstrated in the 2 triangles shown.

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When proving that 2 triangles are congruent, use the diagram and given information to determine whether it will be possible to show that 2 pairs of corresponding angles are congruent or that 2 pairs of corresponding sides are congruent. Then, check if there is enough information to use ASA congruence or SAS congruence.

Practice Problems

1. What triangle congruence theorem could you use to prove triangle ∆𝐴𝐷𝐸 ≅ ∆𝐶𝐵𝐸?

206 | Unit 3

Given Information

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2. Triangle 𝐻𝑀𝐶 and 𝐾𝑊𝑅 are shown. Determine whether the given information is sufficient to prove that the triangles are congruent. If so, state the appropriate congruence theorem or postulate to support that.

Congruent?

Theorem or Postulate Used

∠𝐶𝑀𝐻 ≅ ∠𝑅𝑊𝐾, ∠𝑀𝐻𝐶 ≅ ∠𝑊𝐾𝑅, 𝐻𝑀 ≅ 𝐾𝑊 𝐻𝐶 ≅ 𝐾𝑅, 𝐻𝑀 ≅ 𝐾𝑊, ∠𝑀𝐻𝐶 ≅ ∠𝑊𝐾𝑅

𝐻𝐶 ≅ 𝐾𝑅, 𝐻𝑀 ≅ 𝐾𝑊, ∠𝐻𝐶𝑀 ≅ ∠𝐾𝑊𝑅

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3. Segment 𝐺𝐸 is an angle bisector of both ∠𝐻𝐸𝐹 and ∠𝐹𝐺𝐻. Prove ∆𝐻𝐺𝐸 ≅ ∆𝐹𝐺𝐸.

Unit 3 | 207

Review Problems

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4. Triangles 𝐴𝐶𝐷 and 𝐵𝐶𝐷 are isosceles. Angle 𝐵𝐴𝐶 has a measure of 33° and ∠𝐵𝐷𝐶 has a measure of 35°. Find the measure of ∠𝐴𝐵𝐷.

5. Match each statement using only the information shown in the pairs of congruent triangles. A. The 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle.

C. In the 2 triangles there are 3 pairs of congruent sides.

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B. The 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle.

208 | Unit 3

Unit 3, Lesson 6: Side-Side-Side Triangle Congruence

Warm-Up: Dare to Be Different Construct a triangle with the given side lengths using technology. Side lengths:

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• 2 centimeters (cm) • 1.5 cm • 2.4 cm

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Can you make a triangle that doesn’t look like anyone else’s?

Unit 3 | 209

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Exploration Activity: Proving the Side-Side-Side Triangle Congruence Theorem

Priya was given this task to complete:

Use a sequence of rigid motions to take ∆𝑆𝑇𝑈 onto ∆𝐺𝐻𝐽 given that 𝑆𝑇 ≅ 𝐺𝐻, 𝑇𝑈 ≅ 𝐻𝐽, and 𝑆𝑈 ≅ 𝐺𝐽. For each step, explain how you know that one or more vertices will line up. Help her finish the missing steps in her proof.

motion that takes 𝑆𝑇 to

, so they are congruent. Therefore, there is a rigid

1. 𝑆𝑇 is the same length as

, and the

2. Apply this rigid motion to ∆𝑆𝑇𝑈. The image of 𝑇 will coincide with image of 𝑆 will coincide with

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3. We cannot be sure that the image of 𝑈, which we will call 𝑈′, coincides with

yet. If it does, then our rigid motion takes ∆𝑆𝑇𝑈 to ∆𝐺𝐻𝐽, proving that ∆𝑆𝑇𝑈 ≅ ∆𝐺𝐻𝐽. If it does not, then we continue as follows.

4. 𝐻𝐽 is congruent to the image of

, because rigid motions preserve distance.

5. Therefore, 𝐻 is equidistant from 𝑈′ and

6. A similar argument shows that 𝐺 is equidistant from 𝑈′ and 7. 𝐺𝐻 is the

of the segment connecting 𝑈′ and 𝐽, because the

determined by 2 points that are both equidistant from the endpoints of a segment.

8. Reflection across the

of 𝑈′𝐽, takes

9. Therefore, after the reflection, all 3 pairs of vertices coincide, proving triangles and

210 | Unit 3

are congruent.

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Now, help Priya by finishing a few-sentence summary of her proof. “To prove 2 triangles must be congruent if all 3 pairs of corresponding sides are congruent . . .”

Collaborative Activity: What Else Can Be Concluded about Parallelograms?

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Quadrilateral 𝐴𝐵𝐶𝐷 is a parallelogram. By definition, that means 𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐷 ∥ 𝐶𝐷. Prove that ∆𝐴𝐶𝐷 ≅ ∆𝐶𝐴𝐵 using the side-side-side congruence theorem to show that ∠𝐵 ≅ ∠𝐷.

1. Work on your own to make a diagram, and write a rough draft of a proof.

Unit 3 | 211

2. With your partner, discuss each other’s drafts. a. What do you notice your partner understands about the problem? b. What revision would help them move forward?

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3. Work together to revise your drafts into a clear proof that everyone in your class can follow and agree with.

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Lesson Summary

Side-angle-side (SAS) and angle-side-angle (ASA) congruence conditions can be used to prove triangles congruent using specific conditions involving at least 1 pair of corresponding angles.

Sometimes, however, there is no information given about corresponding pairs of angle measures in triangles. In such cases, the side-side-side (SSS) triangle congruence theorem could be used to prove 2 triangles congruent, where if all 3 pairs of corresponding sides are congruent, then the triangles must be congruent.

To prove that 2 triangles are congruent, look at the diagram and given information, and think about whether it will be easier to find pairs of corresponding angles that are congruent or pairs of corresponding sides that are congruent. Then, check to see if all the information matches ASA, SAS, or SSS congruence conditions. 212 | Unit 3

Practice Problems

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1. A kite is a quadrilateral which has 2 sides next to each other that are congruent and where the other 2 sides are also congruent. Given kite 𝑊𝑋𝑌𝑍, show that at least one of the diagonals of a kite decomposes the kite into 2 congruent triangles.

2. Mai has proven that ∆𝑊𝑌𝑍 is congruent to ∆𝑊𝑌𝑋 using the Side-SideSide Triangle Congruence Theorem. Why can she now conclude that diagonal 𝑊𝑌 bisects ∠𝑍𝑊𝑋 and ∠𝑍𝑌𝑋?

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3. Triangles 𝐵𝐹𝑀 and 𝑊𝑅𝐻 are shown.

Determine whether the given information is sufficient to prove that the triangles are congruent. If so, state the appropriate congruence theorem or postulate to support that.

Given Information

Congruent?

Theorem or Postulate Used

𝐹𝐵 ≅ 𝐻𝑊, 𝐹𝑀 ≅ 𝑅𝑊, ∠𝐵𝐹𝑀 ≅ ∠𝐻𝑊𝑅 𝐹𝐵 ≅ 𝑅𝐻, ∠𝐹𝐵𝑀 ≅ ∠𝑅𝐻𝑊, 𝑀𝐵 ≅ 𝑊𝐻 𝐹𝐵 ≅ 𝐻𝑊, 𝐵𝑀 ≅ 𝐻𝑅, 𝐹𝑀 ≅ 𝑅𝑊

Unit 3 | 213

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4. 𝑊𝑋𝑌𝑍 is a kite. Angle 𝑊𝑋𝑌 has a measure of 133° and ∠𝑍𝑊𝑋 has a measure of 60°. Find the measure of ∠𝑍𝑌𝑊.

Review Problems

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5. Triangles 𝐴𝐶𝐷 and 𝐵𝐶𝐷 are isosceles. Angle 𝐷𝐵𝐶 has a measure of 84° and ∠𝐵𝐷𝐴 has a measure of 24°. Find the measure of ∠𝐵𝐴𝐶. 𝐴𝐷 ≅ 𝐴𝐶 and 𝐵𝐷 ≅ 𝐵𝐶

6. Reflect right triangle 𝐴𝐵𝐶 across 𝐴𝐵. Classify ∆𝐶𝐴𝐶′ according to its side lengths. Explain how you know.

214 | Unit 3

Unit 3, Lesson 7: Angle-Angle-Side and Hypotenuse-Leg Congruence

Warm-Up: Congruence Criteria

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Triangles 𝐴𝐵𝐶 and 𝐾𝑀𝑁 are shown, where 𝐴𝐶 ≅ 𝐾𝑁, 𝐴𝐵 ≅ 𝐾𝑀, and ∠𝐴𝐵𝐶 ≅ ∠𝐾𝑀𝑁.

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2. What do you wonder?

1. What do you notice?

3. Explain whether there is a sequence of rigid motions that could prove ∆𝐴𝐵𝐶 ≅ ∆𝐾𝑀𝑁.

Unit 3 | 215

Exploration Activity: Which Other Criteria Is Enough?

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Five angles (labeled 1 through 5) and 3 line segments (labeled 𝑎, 𝑏, and 𝑐) are given.

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According to the criteria given in each problem, create triangles by choosing the parts needed and copying them onto tracing paper. Helpful hint: When copying an angle, include some (or all) of the dotted lines so that angles and line segments can be connected. 1. Criteria for creating triangles are given in each table.

a. Create 2 different triangles on a separate paper using the steps provided. Criteria 1: Given 2 Angles (AA)

Step 1. Copy ∠1 onto a piece of tracing paper. Step 2. Close the triangle using ∠2.

Step 3. Trace the completed triangle. Label ∠1 and ∠2 in the completed triangle. Step 4. Label your triangle “Criteria 1 (AA).” Step 5. Repeat these steps with ∠3 and ∠5. 216 | Unit 3

b. Create 2 different triangles on a separate paper using the steps provided. Criteria 2: Given 2 Angles and a Nonincluded Side (AAS) Step 1. Copy side 𝑏 onto a piece of tracing paper. Label side 𝑏 on your tracing paper.

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Step 2. Place ∠3 at 1 of the ends of side 𝑏, and trace the ray that does not overlap with side 𝑏 to copy the angle. Label ∠3. Two examples are shown.

Step 3. Use ∠4 to close the triangle such that ∠4 is opposite from side 𝑏. Label ∠4. Step 4. Trace the completed triangle. Label your triangle “Criteria 2 (AAS).” Step 5. Repeat these steps with side 𝑐, ∠1, and ∠2.

2. Compare your 4 triangles from the previous questions with your partner’s.

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3. Write your partner’s name on the line in the table. Then, write “true” or “false” to indicate if your triangle is congruent to your partner’s triangle for each criterion.

Criteria

’s triangle.

∠𝟏 and ∠𝟐

AAS

My triangle is congruent to

∠𝟑 and ∠𝟓

Side 𝒃, ∠𝟑, and ∠𝟒 Side 𝒄, ∠𝟏, and ∠𝟐

Unit 3 | 217

4. With your partner, find another pair of students to form a group of 4. Write the 2 new students’ names in the table. Then, write “true” or “false” to indicate whether your triangle is congruent to your new group members’ triangles for each criterion.

Criteria

My triangle is congruent to

’s

triangle.

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∠𝟏 and ∠𝟐

∠𝟑 and ∠𝟓

AAS

triangle.

Side 𝒃, ∠𝟑, and ∠𝟒 Side 𝒄, ∠𝟏, and ∠𝟐

Explanation

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Criteria

5. Discuss with your group. If given either set of criteria, can you determine that all the parts of the triangles are congruent? Write your explanation in the table.

Given 2 Angles (AA)

Given 2 Angles and a Nonincluded Side (AAS)

218 | Unit 3

Collaborative Activity: Making Connections

Two students shared their thinking based on the given information.

Mabel

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1. Triangles 𝑄𝑊𝐹 and 𝑁𝑇𝑋 are shown. 𝐹𝑄 ≅ 𝑋𝑁, ∠𝐹𝑄𝑊 ≅ ∠𝑋𝑁𝑇, and ∠𝑄𝑊𝐹 ≅ ∠𝑁𝑇𝑋.

Xavier

Xavier thinks the triangles are congruent because he knows that the third angles have to be congruent, and then there would be a side between 2 angles.

Mabel thinks there is not enough information to determine congruence because the side given is not between the 2 angles given.

With your partner, discuss both students’ reasoning. Then, answer the following.

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a. Explain whether you agree with Mabel or Xavier.

b. What additional information could be added to the argument you agree with to make it more convincing?

Unit 3 | 219

2. Triangles 𝐼𝐻𝐺 and 𝐿𝐾𝐽 are shown. 𝐼𝐻 ≅ 𝐿𝐾, 𝐻𝐺 ≅ 𝐾𝐽, and ∠𝐼𝐺𝐻 and ∠𝐿𝐽𝐾 are right angles. Eli and Kaylah had the following discussion.

Eli said, “In a situation where you have 2 sides without the angle in between them, you can’t tell if the triangles are congruent.”

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Kaylah said, “Normally, I would agree with you, except these are right triangles.” a. Discuss with your partner what Kaylah might be thinking.

b. Given 2 sides of a right triangle, how can you determine the length of the third side?

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Kaylah continued, “For a right triangle, as long as I have 2 of the sides, I can find the length of the third side. Then, I can use SSS since all 3 corresponding sides would be congruent.”

c. Work with your partner to explain why Kaylah’s argument could work to show congruence for any pair of right triangles where the hypotenuse and 1 leg are identified as congruent.

d. Explain whether Kaylah’s argument would be needed to show congruence between right triangles where both legs of the triangles are identified as congruent.

220 | Unit 3

Guided Activity: Triangle Congruence

2. Complete the statements.

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1. Explain how the angle-angle-side (AAS) congruence conditions are sufficient to prove triangles are congruent. You can use the triangles shown as an example.

If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the similar. congruent.

If a congruent side opposite 1 of the

third pair of angles are

also applies. Thus, AAS is enough to show congruence.

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SSS SAS ASA

given angles is congruent to the corresponding side of the other triangle, then

When 2 corresponding sides and nonincluded corresponding angles of 2 triangles are congruent, that is not sufficient to prove the triangles are congruent. 3. Suppose 2 right triangles each have the hypotenuse and 1 leg that are congruent. Explain why this is sufficient information to prove congruence for right triangles, despite using 2 sides with a nonincluded angle.

Unit 3 | 221

Lesson Summary Side-angle-side (SAS), angle-side-angle (ASA), and side-side-side (SSS) triangle congruence conditions were explored in previous lessons. Two additional congruence conditions were explored in this lesson.

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• Angle-angle-side (AAS) triangle congruence theorem: If 2 angles and a side opposite of 1 of them are congruent to 2 angles and the corresponding side of another triangle, then the triangles are congruent. • Hypotenuse-leg (HL) theorem: Two right triangles are congruent if the hypotenuse and a leg of 1 triangle are congruent to the hypotenuse and a leg of the other.

Practice Problems

Any of these congruence conditions can be used to prove 2 triangles are congruent. When proving triangles congruent, use the given information to determine which conditions can be used.

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1. Triangles 𝐻𝐾𝐶 and 𝐷𝑇𝑅 are shown. Determine if the given information is sufficient to justify that the triangles are congruent. If so, name the congruence theorem or postulate that supports that.

Given Information

Are They Congruent?

Theorem or Postulate Used

𝐻𝐾 ≅ 𝐷𝑇, 𝐾𝐶 ≅ 𝑇𝑅, ∠𝐾𝐶𝐻 ≅ ∠𝑇𝑅𝐷

∠𝐾𝐶𝐻 ≅ ∠𝑇𝑅𝐷, ∠𝐻𝐾𝐶 ≅ ∠𝐷𝑇𝑅, ∠𝐶𝐻𝐾 ≅ ∠𝑅𝐷𝑇 ∠𝐾𝐶𝐻 ≅ ∠𝑇𝑅𝐷, ∠𝐻𝐾𝐶 ≅ ∠𝐷𝑇𝑅, 𝐻𝐶 ≅ 𝐷𝑅 𝐻𝐾 ≅ 𝐷𝑇, 𝐾𝐶 ≅ 𝑇𝑅, 𝐻𝐾 ⊥ 𝐻𝐶, 𝐷𝑇 ⊥ 𝐷𝑅 𝐻𝐾 ≅ 𝐷𝑇, 𝐻𝐶 ≅ 𝐷𝑅, ∠𝐾𝐻𝐶 ≅ ∠𝑇𝐷𝑅

222 | Unit 3

2. Which of the following criteria always proves triangles congruent? Select all that apply.

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□ 3 congruent angles □ 3 congruent sides □ Corresponding congruent Side-Angle-Side □ Corresponding congruent Side-Side-Angle □ Corresponding congruent Angle-Side-Angle

Review Problems

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3. Jada states that diagonal 𝑊𝑌 bisects angles 𝑍𝑊𝑋 and 𝑍𝑌𝑋. Is she correct? Explain your reasoning.

4. Select all true statements based on the diagram.

□ Angle 𝐶𝐵𝐸 is congruent to angle 𝐷𝐴𝐸. □ Angle 𝐶𝐸𝐵 is congruent to angle 𝐷𝐸𝐴. □ Segment 𝐷𝐴 is congruent to segment 𝐶𝐵. □ Segment 𝐷𝐶 is congruent to segment 𝐴𝐵. □ Line 𝐷𝐶 is parallel to line 𝐴𝐵. □ Line 𝐷𝐴 is parallel to line 𝐶𝐵.

Unit 3 | 223

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Unit 3, Lesson 8: Definition of Congruence in Terms of Rigid Motions

Warm-Up: Brain Teaser Brain teasers like this have been traced back to medieval times.

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1. A man has to take a panther, a raccoon, and some strawberries across the river. His rowboat has only enough room for the man plus the panther, the raccoon, or the strawberries. If he takes the strawberries with him, the panther will eat the raccoon. If he takes the panther, the raccoon will eat the strawberries. Only when the man is present are the raccoon and the strawberries safe from their enemies. All the same, the man successfully carries the panther, the raccoon, and the strawberries across the river. How?

Unit 3 | 225

Collaborative Activity: Justifying Congruence 1. Triangles 𝑇𝐶𝑀 and 𝑅𝐺𝑁 are shown, where 𝑇𝑀 ≅ 𝑅𝑁, 𝑀𝐶 ≅ 𝑁𝐺, and ∠𝑇𝑀𝐶 ≅ ∠𝑅𝑁𝐺.

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a. Discuss with your partner what can be concluded from the given information.

b. Ask a classmate for 2 conclusions they made. Record their conclusions, summarize their reasons. You are the only person who should write in the first 2 columns of the table. Have your classmate initial next to your summary, indicating that your summary is correct.

My Summary of Their Reason

Initials

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Conclusion

c. What rigid motions can be applied so that ∆𝑇𝐶𝑀 is mapped onto ∆𝑅𝐺𝑁?

d. Discuss with your partner what can be concluded from the rigid motions.

226 | Unit 3

e. Ask a classmate for 2 conclusions they made. Record their conclusions, and summarize their reasons. You are the only person who should write in the first 2 columns of the table. Have your classmate initial next to your summary, indicating that your summary is correct. My Summary of Their Reason

Initials

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Conclusion

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f. Discuss with your partner why showing that the triangles are congruent using the definition of congruence in terms of rigid transformations shows that SAS congruence is sufficient to prove the triangles are congruent. Summarize your discussion.

Unit 3 | 227

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2. Triangles 𝑉𝑊𝐾 and 𝑆𝐻𝑃 are shown. Assume 𝑉𝐾 ≅ 𝑆𝑃, 𝑊𝑉 ≅ 𝐻𝑆, and 𝐾𝑊 ≅ 𝑃𝐻.

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a. Explain how you could use tracing paper to verify ∆𝑉𝑊𝐾 ≅ ∆𝑆𝐻𝑃.

b. Write a sequence of transformations that will map ∆𝑉𝑊𝐾 onto ∆𝑆𝐻𝑃.

c. Show each transformation on the coordinate grid. • Draw ∆𝑉′𝑊′𝐾′, the result of the first transformation. • Map the vertices of ∆𝑉𝑊𝐾 to ∆𝑉′𝑊′𝐾′. • Map the vertices of ∆𝑉′𝑊′𝐾′ to ∆𝑆𝐻𝑃.

228 | Unit 3

d. Complete the statements. The outcome of mapping the transformations of ∆𝑉𝑊𝐾 to ∆𝑉′𝑊′𝐾′ and then of

𝐻, 𝑆, 𝑃,

shows that 𝑉′ coincides with

and 𝐾′ coincides with

congruence similarity ∆𝑉𝑊𝐾 is

𝐻. 𝑆. 𝑃.

𝐻, 𝑆, 𝑃,

𝑊′ coincides with

Because the definition of

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∆𝑉′𝑊′𝐾′ to ∆

in terms of rigid motions preserves distance and angles,

congruent similar

to ∆

AAS ASA SAS SSS

is sufficient to show

𝑊𝑉 ≅ 𝐻𝑆, and 𝐾𝑊 ≅ 𝑃𝐻 were given,

. Since 𝑉𝐾 ≅ 𝑆𝑃,

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the triangles are congruent.

Unit 3 | 229

3. Triangles 𝑅𝑀𝐵 and 𝐺𝑁𝐶 are shown, where 𝑅𝑀 ≅ 𝐺𝑁, ∠𝐵𝑅𝑀 ≅ ∠𝐶𝐺𝑁, and ∠𝐵𝑀𝑅 ≅ ∠𝐶𝑁𝐺.

Adriana knows that there is a series of rigid motions that maps ∆𝑅𝑀𝐵 onto ∆𝐺𝑁𝐶.

□ Line segment 𝑅𝐵 coincides with 𝐺𝑁.

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a. Select all of the statements that are true after the transformations.

□ Line segment 𝐵𝑀 coincides with 𝐶𝑁. □ Line segment 𝐶𝐺 coincides with 𝐵𝑅. □ Angle 𝑁𝐶𝐺 coincides with ∠𝑅𝑀𝐵. □ Angle 𝑅𝐵𝑀 coincides with ∠𝐺𝐶𝑁.

b. Explain how the given information and the definition of congruence in terms of rigid motions show that ∆𝑅𝑀𝐵 ≅ ∆𝐺𝑁𝐶.

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Lesson Summary

The definition of congruence in terms of rigid motions states that 2 figures are congruent if and only if there exists 1 or more rigid motions that will map 1 figure onto the other.

For example, ∆𝑅𝐶𝐻 and ∆𝑁𝐾𝑇 are shown on the coordinate plane, where ∠𝐻𝑅𝐶 ≅ ∠𝑇𝑁𝐾, ∠𝑅𝐶𝐻 ≅ ∠𝑁𝐾𝑇, and 𝐶𝐻 ≅ 𝐾𝑇. A translation and a reflection can map ∆𝑅𝐶𝐻 onto ∆𝑁𝐾𝑇, so that 𝑅 coincides with 𝑁, 𝐶 coincides with 𝐾, and 𝐻 coincides with 𝑇. Because of the definition of congruence in terms of rigid motions, ∆𝑅𝐶𝐻 ≅ ∆𝑁𝐾𝑇.

Congruence conditions could also be used. Since ∠𝐻𝑅𝐶 ≅ ∠𝑇𝑁𝐾, ∠𝑅𝐶𝐻 ≅ ∠𝑁𝐾𝑇, and 𝐶𝐻 ≅ 𝐾𝑇, AAS congruence is also sufficient to show that the triangles are congruent. 230 | Unit 3

Practice Problems 1. Triangles 𝑊𝑋𝐷 and 𝑅𝑁𝐻 are shown, where 𝑅𝑁 ≅ 𝑋𝐷, 𝐷𝑊 ≅ 𝑁𝐻, and ∠𝑅𝑁𝐻 ≅ ∠𝑋𝐷𝑊.

b. Complete the statements. The outcome of mapping the sequence of transformations

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a. Write a sequence of transformations that will map ∆𝐻𝑅𝑁 onto ∆𝑊𝑋𝐷.

𝐻 coincides with point

, point

, and point 𝑁 coincides with point

𝑅 coincides with point

described in part A shows that point

. Therefore,

because of the definition of congruence in terms of rigid motions, it can be stated . Since 𝑅𝑁 ≅ 𝑋𝐷, 𝐷𝑊 ≅ 𝑁𝐻, and ∠𝑅𝑁𝐻 ≅ ∠𝑋𝐷𝑊 were given,

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that ∆𝐻𝑅𝑁 ≅

congruence is sufficient to show the triangles are congruent.

Unit 3 | 231

2. Triangles 𝑃𝐻𝑁 and 𝑊𝑇𝑅 are shown, where 𝑊𝑇 ≅ 𝐻𝑃, ∠𝑃𝐻𝑁 ≅ ∠𝑇𝑊𝑅, and ∠𝐻𝑁𝑃 ≅ ∠𝑊𝑅𝑇.

□ Line segment 𝐻𝑁 coincides with 𝑊𝑅.

□ Line segment 𝐻𝑁 coincides with 𝑊𝑇.

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a. A series of rigid motions maps ∆𝑃𝐻𝑁 onto ∆𝑇𝑊𝑅. Select all of the statements that are true after the transformations.

□ Line segment 𝑃𝑁 coincides with 𝑇𝑅. □ Angle 𝑊𝑇𝑅 coincides with ∠𝐻𝑃𝑁. □ Line segment 𝑃𝑁 coincides with 𝑊𝑅.

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Review Problems

b. Based on the given information, which triangle congruence conditions are sufficient to show that ∆𝑃𝐻𝑁 maps onto ∆𝑇𝑊𝑅?

3. The triangles are congruent. Which sequence of rigid motions takes ∆𝐷𝐸𝐹 onto ∆𝐵𝐴𝐶?

A. Translate ∆𝐷𝐸𝐹 using directed line segment 𝐸𝐴. Rotate ∆𝐷′𝐸′𝐹′ using 𝐴 as the center so that 𝐷′ coincides with 𝐶. Reflect 𝐷″𝐸″𝐹″ across 𝐴𝐶.

B. Translate ∆𝐷𝐸𝐹 using directed line segment 𝐸𝐴. Rotate ∆𝐷′𝐸′𝐹′ using 𝐴 as the center so that 𝐷′ coincides with 𝐶. Reflect 𝐷″𝐸″𝐹″ across 𝐴𝐵. C. Translate ∆𝐷𝐸𝐹 using directed line segment 𝐸𝐴. Rotate ∆𝐷′𝐸′𝐹′ using 𝐴 as the center so that 𝐷′ coincides with 𝐵. Reflect 𝐷″𝐸″𝐹″ across 𝐴𝐶. D. Translate ∆𝐷𝐸𝐹 using directed line segment 𝐸𝐴. Rotate ∆𝐷′𝐸′𝐹′ using 𝐴 as the center so that 𝐷′ coincides with 𝐵. Reflect 𝐷″𝐸″𝐹″ across 𝐴𝐵.

232 | Unit 3

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4. Triangles 𝐴𝐷𝐸 and 𝐸𝐶𝐵 are shown, where ∠𝐴 ≅ ∠𝐶, 𝐴𝐸 ≅ 𝐶𝐸. Write a proof to show that ∆𝐴𝐷𝐸 ≅ ∆𝐶𝐵𝐸.

Unit 3 | 233

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Unit 3, Lesson 9: Corresponding Parts of Congruent Triangles

Warm-Up: Notice and Wonder: Congruence Fail

In triangles 𝐺𝐵𝐷 and 𝐾𝐻𝐼:

• Angle 𝐺𝐵𝐷 is congruent to angle 𝐾𝐻𝐼.

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What do you notice? What do you wonder?

• Segment 𝐵𝐷 is congruent to segment 𝐻𝐼.

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• Segment 𝐷𝐺 is congruent to segment 𝐼𝐾.

Unit 3 | 235

Exploration Activity: Overlapping Triangles Polygon 𝑋𝐹𝐾𝐵 is shown. Work with your partner to complete the following.

2. Draw ∆𝑋𝐵𝑊 and ∆𝑋𝐹𝑅 separately. Label where point 𝐾 is on each triangle.

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1. How many triangles are in polygon 𝑋𝐹𝐾𝐵?

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3. Discuss with your partner if ∆𝑋𝐵𝑊 and ∆𝑋𝐹𝑅 have any common sides or angles. If so, name the common part(s).

4. Explain why knowing if ∆𝑋𝐵𝑊 and ∆𝑋𝐹𝑅 have any common sides or angles could be helpful in a proof.

5. In your drawing of ∆𝑋𝐵𝑊 and ∆𝑋𝐹𝑅, mark the common part(s) as congruent. 6. What property justifies that something is equal to itself?

236 | Unit 3

7. Mark the given information on your drawing. Given: 𝑅𝑋 ≅ 𝑊𝑋, ∠𝑋𝑅𝐾 ≅ ∠𝑋𝑊𝐾

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8. What can you conclude about ∆𝑋𝐵𝑊 and ∆𝑋𝐹𝑅 from the information you marked on your drawing?

9. Complete the flowchart proof for polygon 𝑋𝐹𝐾𝐵.

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Prove: ∠𝑋𝐵𝐾 ≅ ∠𝑋𝐹𝐾

Given: 𝑅𝑋 ≅ 𝑊𝑋, ∠𝑋𝑅𝐾 ≅ ∠𝑋𝑊𝐾

10. Check your proof with another group, and adjust your proof if necessary.

Unit 3 | 237

Guided Activity: Proofs Using CPCTC Once 2 triangles have been proven to be congruent, then Corresponding Parts of Congruent Triangles Are Congruent (CPCTC) states that any other corresponding parts of the triangles are also congruent. In the flowchart proof, proving that ∠𝑋𝐵𝐾 ≅ ∠𝑋𝐹𝐾 first required proving that ∆𝑋𝐵𝑊 and ∆𝑋𝐹𝑅 are congruent.

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1. Triangles 𝑇𝑌𝑅 and 𝑋𝑌𝑊 are shown. a. Mark the diagram to show the following given information.

Given: Point 𝑌 is the midpoint of 𝑊𝑇. Point 𝑌 is the midpoint of 𝑅𝑋.

b. What does “point 𝑌 is the midpoint of 𝑊𝑇” allow you to conclude about the diagram?

c. What does “point 𝑌 is the midpoint of 𝑅𝑋” allow you to conclude about the diagram?

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d. Discuss with your partner whether there is any other information about the 2 triangles that you can find from the diagram. Summarize your discussion.

e. Work with your partner to complete the paragraph proof. Given: Point 𝑌 is the midpoint of 𝑊𝑇. Point 𝑌 is the midpoint of 𝑅𝑋.

Prove: 𝑇𝑅 ≅ 𝑋𝑊

It is given that 𝑌 is the midpoint of 𝑊𝑇, so midpoint of 𝑅𝑋, so

by . Also, it is given that 𝑌 is the

. Therefore, by ,∆

238 | Unit 3

≅∆

, so by

, 𝑇𝑅 ≅ 𝑋𝑊.

2. Triangles 𝐶𝑋𝐻 and 𝑋𝑀𝑉 are shown. Given: 𝐶𝑉 ⊥ 𝐻𝑀, 𝐻𝑋 ≅ 𝑋𝑉, 𝐻𝐶 ≅ 𝑀𝑉 Prove: 𝑋𝐶 ≅ 𝑋𝑀

b. Which congruence theorem or postulate do you think should be used?

1. 𝐶𝑉 ⊥ 𝐻𝑀

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Reason

1. Given

Statement

c. Complete the two-column proof.

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a. Mark the diagram to show the given information.

2. 3.

7. 𝑋𝐶 ≅ 𝑋𝑀

Unit 3 | 239

Collaborative Activity: Proofs Using CPCTC 1. A diagram of triangles 𝑁𝐷𝐻 and 𝑃𝐾𝐹 with 𝑁𝐾 is shown. Given: 𝐻𝐷 ∥ 𝐹𝑃, 𝐻𝐷 ≅ 𝐹𝑃, 𝐻𝑁 ≅ 𝐾𝑃

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Prove: ∠𝐻𝑁𝐷 ≅ ∠𝑃𝐾𝐹

Complete the paragraph proof. It is given that 𝐻𝐷 ∥ 𝐹𝑃, so by ≅∠

. Since it is given that 𝐻𝐷 ≅ 𝐹𝑃 and 𝐻𝑁 ≅ 𝐾𝑃, ∆

. Therefore, ∠𝐻𝑁𝐷 ≅ ∠𝑃𝐾𝐹 by

Lesson Summary

≅∆

∠

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Sometimes the goal of a proof is to show a pair of corresponding angles or sides in triangles are congruent. In such cases, it may be necessary to first prove the triangles are congruent using SAS, ASA, SSS, AAS, or HL congruence conditions. Once it’s proven that 2 triangles are congruent, then corresponding parts of congruent triangles are congruent (CPCTC) can be used to state the congruence of other corresponding parts of the triangles. CPCTC states that if 2 triangles are congruent, then all corresponding parts of the triangles are congruent.

For example, ∆𝑃𝑀𝐿 and ∆𝑃𝑊𝐿 are shown, where ∠𝑀𝑃𝐿 ≅ ∠𝑊𝑃𝐿 and ∠𝑀𝐿𝑃 ≅ ∠𝑊𝐿𝑃. Since 𝑃𝐿 ≅ 𝑃𝐿 by the reflexive property of congruence, ∆𝑃𝑀𝐿 ≅ ∆𝑃𝑊𝐿 by ASA congruence. Since ∆𝑃𝑀𝐿 ≅ ∆𝑃𝑊𝐿, by CPCTC, 𝑀𝐿 ≅ 𝑊𝐿, 𝑃𝑀 ≅ 𝑃𝑊, and ∠𝐿𝑀𝑃 ≅ ∠𝐿𝑊𝑃.

240 | Unit 3

Practice Problems 1. Quadrilateral 𝑀𝐻𝑅𝐷 is shown.

Given: 𝐷𝑀 ∥ 𝑅𝐻, ∠𝑅𝐷𝑀 ≅ ∠𝑅𝐻𝑀

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Prove: 𝐷𝑀 ≅ 𝑅𝐻

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Complete the flowchart proof.

Unit 3 | 241

2. Triangle 𝑉𝑅𝑈 is shown.

Given: 𝑈𝑅 ≅ 𝑉𝑅 and 𝑅𝑇 ⊥ 𝑈𝑉 Prove: ∠𝑉𝑅𝑇 ≅ ∠𝑈𝑅𝑇

Statement

Reason Given

𝑈𝑅 ≅ 𝑉𝑅

Given

𝑅𝑇 ⊥ 𝑈𝑉

Definition of perpendicular lines

∆𝑇𝑉𝑅 ≅ ∆𝑇𝑈𝑅

∆𝑉𝑇𝑅 and ∆𝑇𝑅𝑈 are right triangles. 𝑇𝑅 ≅ 𝑇𝑅

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Complete the two-column proof.

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∠𝑉𝑅𝑇 ≅ ∠𝑈𝑅𝑇

Review Problems

3. Is triangle 𝐴𝐹𝐸 congruent to triangle 𝐴𝐷𝐸?

Explain your reasoning.

242 | Unit 3

𝐴𝐹 ≅ 𝐴𝐷, ∠𝐹 ≅ ∠𝐷

4. Triangle 𝐷𝐴𝐶 is isosceles with congruent sides 𝐴𝐷 and 𝐴𝐶. Which additional given information is sufficient for showing that ∆𝐷𝐵𝐶 is isosceles? Select all that apply.

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□ Segment 𝐷𝐵 is congruent to 𝐵𝐶. □ Segment 𝐴𝐵 is congruent to 𝐵𝐷. □ Angle 𝐴𝐵𝐷 is congruent to ∠𝐴𝐵𝐶. □ Angle 𝐴𝐷𝐶 is congruent to ∠𝐴𝐶𝐷. □ Segment 𝐴𝐵 is an angle bisector of ∠𝐷𝐴𝐶. □ Triangle 𝐵𝐷𝐴 is congruent to ∆𝐵𝐷𝐶.

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5. In the image, triangle 𝐴𝐵𝐶 is congruent to triangle 𝐵𝐴𝐷 and triangle 𝐶𝐸𝐴. What are the measures of the 3 angles in triangle 𝐶𝐸𝐴? Show or explain your reasoning.

Unit 3 | 243

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Unit 3, Lesson 10: Practicing Proofs

Warm-Up: Which One Doesn’t Belong: Intersecting Lines

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Which one doesn’t belong?

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Unit 3 | 245

Exploration Activity: Perpendicular Bisector Diego, Jada, and Noah were given the following task: Prove that if a point 𝐶 is the same distance from 𝐴 as it is from 𝐵, then 𝐶 must be on the perpendicular bisector of 𝐴𝐵.

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At first they were really stuck. Noah asked, “How do you prove a point is on a line?” Their teacher gave them the hint, “Another way to think about it is to draw a line that you know 𝐶 is on, and prove that line has to be the perpendicular bisector.” They each drew a line and thought about their pictures. Here are their rough drafts.

Diego’s approach: “I drew a line through 𝐶 that was perpendicular to 𝐴𝐵 and through the midpoint of 𝐴𝐵. That line is the perpendicular bisector of 𝐴𝐵 and 𝐶 is on it, so that proves 𝐶 is on the perpendicular bisector.”

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Jada’s approach: “I thought the line through 𝐶 would probably go through the midpoint of 𝐴𝐵 so I drew that and labeled the midpoint 𝐷. Triangle 𝐴𝐶𝐵 is isosceles, so ∠𝐴 and ∠𝐵 are congruent, and 𝐴𝐶 and 𝐵𝐶 are congruent. And 𝐴𝐷 and 𝐷𝐵 are congruent because 𝐷 is a midpoint. That made two congruent triangles by the Side-AngleSide Triangle Congruence Theorem. So I know ∠𝐴𝐷𝐶 and ∠𝐵𝐷𝐶 are congruent, but I still don’t know if 𝐷𝐶 is the perpendicular bisector of 𝐴𝐵.”

Noah’s approach: “In the Isosceles Triangle Theorem proof, Mai and Kiran drew an angle bisector in their isosceles triangle, so I’ll try that. I’ll draw the angle bisector of ∠𝐴𝐶𝐵. The point where the angle bisector hits 𝐴𝐵 will be 𝐷. So ∆𝐴𝐶𝐷 and ∆𝐵𝐶𝐷 are congruent, which means 𝐴𝐷 and 𝐵𝐷 are congruent, so 𝐷 is a midpoint and 𝐶𝐷 is the perpendicular bisector.” 1. With your partner, discuss each student’s approach. • What do you notice that this student understands about the problem? • What question would you ask them to help them move forward? 246 | Unit 3

2. Using the ideas you heard and the ways you think each student could make their explanation better, write your own explanation for why 𝐶 must be on the perpendicular bisector of 𝐴 and 𝐵.

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Collaborative Activity: It’s in the Bag! With your group, complete each proof using the statement and reason cards provided. Proof A Given: ∠𝑌𝑆𝐷 ≅ ∠𝐹𝑀𝐷

Prove: 𝐹𝑀 ≅ 𝑌𝑆

Reason

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Statement

∆𝑌𝐷𝐹 is isosceles, with base 𝑌𝐹.

Unit 3 | 247

Proof B Given: 𝑇𝑁 bisects 𝐻𝑇𝑋. 𝐻𝑇 ≅ 𝑋𝑇

Prove: 𝑇𝑁 bisects ∠𝐻𝑁𝑋. Reason

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Statement

248 | Unit 3

Proof C Given: ∠𝑊𝑃𝑇 ≅ ∠𝐾𝐹𝑀 𝑃𝑊 ≅ 𝐹𝐾

∠𝑌𝑀𝑇 ≅ ∠𝑌𝑇𝑀

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Prove: ∆𝑃𝑊𝑇 ≅ ∆𝐹𝐾𝑀

Reason

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Statement

Unit 3 | 249

Proof D Given: 𝐿𝑀 ≅ 𝐿𝑅 𝐿𝑅 ⊥ 𝑅𝑍

𝐿𝑀 ⊥ 𝑀𝑍

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Prove: ∠𝑀𝑍𝐿 ≅ ∠𝑅𝑍𝐿

Reason

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Statement

250 | Unit 3

Proof E Given: 𝑅𝐹 ≅ 𝑆𝑊

∠𝑇𝑆𝐹 ≅ ∠𝑀𝑊𝑅

𝐹𝑇 ∥ 𝑀𝑅

Prove: ∆𝑇𝑆𝐹 ≅ ∆𝑀𝑊𝑅

Reason

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Statement

Unit 3 | 251

Proof F Given: 𝑃𝐹 ≅ 𝑊𝑅

𝑃𝑊 ≅ 𝐹𝑅

Prove: 𝑃𝐹 ∥ 𝑊𝑅

Reason

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Statement

Lesson Summary

To prove that segments or angles are congruent, look for triangles that those segments or angles are part of. Can the triangles be proven congruent? Are the segments or angles corresponding parts of congruent triangles? Does that help prove the conjecture? To prove that the triangles are congruent, look at the diagram and given information. Think about whether it will be easier to find pairs of corresponding angles that are congruent or pairs of corresponding sides that are congruent. Then, check if there’s enough information to use the SSS, SAS, ASA, AAS, or HL congruence conditions. Apply the fact that corresponding parts of congruent figures must be congruent (CPCTC) to prove congruent corresponding parts. 252 | Unit 3

Practice Problems

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1. In isosceles triangle 𝐷𝐴𝐶, 𝐴𝐷 is congruent to 𝐴𝐶. Kiran knows that the base angles of an isosceles triangle are congruent. What additional information does Kiran need to know in order to show that 𝐴𝐵 is a perpendicular bisector of segment 𝐶𝐷?

2. Leighton wrote a proof to show that ∆𝐻𝐸𝐺 ≅ ∆𝐹𝐸𝐺, but she made an error in her proof. Leighton’s proof is shown. Given: 𝐸𝐺 bisects ∠𝐹𝐺𝐻.

Prove: ∆𝐻𝐸𝐺 ≅ ∆𝐹𝐸𝐺

Reason

𝐸𝐺 bisects ∠𝐹𝐺𝐻.

Given

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Statement

∠𝐸𝐺𝐻 ≅ ∠𝐸𝐺𝐹

Definition of angle bisector

𝐸𝐺 ≅ 𝐸𝐺

Reflexive property of congruence

∠𝐻𝐸𝐺 ≅ ∠𝐹𝐸𝐺 ∆𝐻𝐸𝐺 ≅ ∆𝐹𝐸𝐺

Definition of angle bisector

ASA congruence

Describe the error Leighton made in her proof.

Unit 3 | 253

3. Triangles 𝑂𝐵𝑊 and 𝑊𝑇𝑌 are shown. Given: ∠𝑂𝐵𝑊 ≅ ∠𝑇𝑌𝑊, 𝑂𝑊 ≅ 𝑇𝑊 Prove: 𝑂𝐵 ≅ 𝑇𝑌

Complete the proof.

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It is given that ∠𝑂𝐵𝑊 ≅ ∠𝑇𝑌𝑊 and 𝑂𝑊 ≅ 𝑇𝑊. ∠𝑂𝑊𝐵 ≅ ∠𝑇𝑊𝑌 because

. Therefore, ∆𝑂𝐵𝑊 ≅ ∆𝑇𝑌𝑊

. By

Review Problems

, 𝑂𝐵 ≅ 𝑇𝑌.

𝐴𝐷 ≅ 𝐴𝐶

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4. Triangle 𝐷𝐴𝐶 is isosceles. What information do you need to show that ∆𝐷𝐵𝐴 is congruent to ∆𝐶𝐵𝐴 by the Side-Angle-Side Triangle Congruence Theorem?

5. 𝑊𝑋𝑌𝑍 is a kite. Angle 𝑊𝑋𝑌 has a measure of 133° and angle 𝑍𝑌𝑋 has a measure of 34°. Find the measure of angle 𝑍𝑊𝑌.

254 | Unit 3

Unit 3, Lesson 11: Solving Problems Using Congruence

Warm-Up: Relationships in Congruent Triangles

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Triangles 𝑅𝐶𝑊 and 𝐾𝐻𝐶 are shown.

Prove: ∠𝑊𝑅𝐶 ≅ ∠𝐶𝐾𝐻

Given: 𝑅𝑊 ∥ 𝐾𝐶, 𝐶 is the midpoint of 𝑊𝐻, 𝑊𝑅 ≅ 𝐶𝐾

1. Complete the missing statements and reasons in the two-column proof.

Statement

Reason

1. Given

2. Definition of midpoint

3. 𝑅𝑊 ∥ 𝐾𝐶

3. Given

5. Given

7. ∠𝑊𝑅𝐶 ≅ ∠𝐶𝐾𝐻

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1. 𝐶 is the midpoint of 𝑊𝐻.

Unit 3 | 255

Collaborative Activity: Using CPCTC to Determine Measures Triangle 𝑀𝐵𝐶 is shown, where 𝑀𝑋 = 3.6, 𝑀𝐵 = 7.2, 𝑅𝑋 = 2, 𝐵𝐶 = 6.5, 𝑅𝐵 = 3.9, 𝑚∠𝑌𝐵𝑅 = 29°, 𝑚∠𝐶𝑋𝑅 = 83°, and 𝑚∠𝑀𝐶𝐵 = 63°.

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Work with your partner to complete the following.

1. How many different triangles are in ∆𝑀𝐵𝐶?

𝑚∠𝑅𝑌𝐵 =

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𝑅𝐶 =

2. If ∆𝐵𝑅𝑌 ≅ ∆𝐶𝑅𝑋, then determine the following measures.

3. If ∆𝐵𝑋𝑀 ≅ ∆𝐶𝑌𝑀, then determine the following measures. 𝐶𝑌 =

256 | Unit 3

𝐶𝑀 =

𝑚∠𝑀𝐶𝑌 =

Collaborative Activity: Corresponding Parts of Congruent Polygons 1. The diagram shows ∆𝐻𝐾𝐶 and ∆𝑇𝐷𝑅, where ∆𝐻𝐾𝐶 ≅ ∆𝐷𝑇𝑅. Some measurements of the sides are given. 𝐾𝐻 = 5𝑦 − 2, 𝐷𝑅 = 6𝑦 − 5, and 𝑇𝐷 = 4𝑦 + 3.

Arthur’s Work 5𝑦 − 2 = 6𝑦 − 5 −2 = 𝑦 − 5 3=𝑦

𝐾𝐻 = 13

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a. Find Arthur’s error.

𝐾𝐻 = 5(3) − 2

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Arthur used the information to determine the length of 𝐾𝐻. His work is shown.

b. Help Arthur understand and learn from his mistake. Write a brief note to Arthur explaining what his mistake was and how he can learn from it.

Unit 3 | 257

2. Quadrilateral 𝑌𝐷𝑊𝑇 is shown, where 𝑌𝐻 is a perpendicular bisector of 𝑇𝐷.

a. How many different triangles are in quadrilateral 𝑌𝐷𝑊𝑇?

𝐻𝐷 =

𝑚∠𝐷𝑌𝐻 =

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b. If ∆𝐷𝐻𝑌 ≅ ∆𝑇𝐻𝑌, 𝑌𝐷 = 130, 𝑌𝐻 = 7, and 𝑚∠𝐻𝑇𝑌 = 38°, determine the following measures.

𝑇𝐻 =

𝑚∠𝐻𝑇𝑊 =

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𝑚∠𝑇𝑊𝐻 =

c. If ∆𝐷𝐻𝑊 ≅ ∆𝑇𝐻𝑊, 𝑊𝐻 = 5, and 𝑚∠𝐻𝑊𝐷 = 61°, use these and all other known measurements to determine the following measures.

𝑇𝑊 =

Lesson Summary

If a triangle can be mapped onto another triangle using rigid transformations, all the parts of the triangles are congruent. This understanding can then be applied to determine other measurements in the 2 congruent triangles. Congruent triangles have equal corresponding side lengths and equal corresponding angle measures. To find measurements of side lengths or angle measures of 2 congruent triangles, create equations where the values or expressions of the corresponding parts are set equal to each other. Solve for the variables. Remember to substitute the value of the variable back into the expression, if needed, to find the length of an unknown side or the measure of an unknown angle. 258 | Unit 3

Practice Problems 1. Triangles 𝑇𝐿𝐶 and 𝑅𝑁𝐾 are shown, where ∆𝑇𝐿𝐶 ≅ ∆𝑁𝐾𝑅, 𝑚∠𝑇𝐶𝐿 = 95°, 𝑚∠𝐶𝐿𝑇 = (6𝑥 + 11)°, 𝑚∠𝑁𝐾𝑅 = (10𝑥 − 7)°, and 𝑚∠𝑁𝑅𝐾 = (13𝑦 + 4)°.

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a. Determine the value of 𝑥.

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c. Find the measure of ∠𝐶𝑇𝐿.

b. Determine the value of 𝑦.

2. Triangles 𝐶𝑋𝐻 and 𝑋𝑀𝑉 are shown, where ∆𝐻𝐶𝑋 ≅ ∆𝑉𝑀𝑋, 𝐶𝑋 = 2𝑤 + 16, 𝐻𝑋 = 26, 𝐶𝐻 = 5𝑤 − 9, 𝑀𝑋 = 4.5𝑤 − 11.5, and 𝑉𝑋 = 4𝑡 + 6.

a. Determine the value of 𝑡.

b. Determine the value of 𝑤.

Unit 3 | 259

3. Triangles 𝑋𝐻𝑇 and 𝐷𝑅𝑊 are shown, where ∆𝐻𝑇𝑋 ≅ ∆𝐷𝑊𝑅, 𝐻𝑋 = 5𝑦 + 0.5, 𝐻𝑇 = 11𝑦 − 4.2, and 𝐷𝑅 = 7𝑦 − 3.9. Penelope used this information to determine the length of 𝐷𝑊, but she made an error in her work. Penelope’s work is shown.

7𝑦 − 3.9 = 11𝑦 − 4.2 0.3 = 4𝑦

𝑦 = 0.075

𝐷𝑊 = 𝐻𝑇 = 11(0.075) − 4.2 = −3.375

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Penelope’s Work

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b. Find the length of 𝐷𝑊.

a. Describe the error Penelope made in her work.

Review Problems

4. When ∆𝐴𝐵𝐶 is reflected across 𝐴𝐵, the image is ∆𝐴𝐵𝐷. Why are 𝐴𝐷 and 𝐴𝐶 congruent? A. Congruent parts of congruent figures are corresponding. B. Corresponding parts of congruent figures are congruent. C. An isosceles triangle has a pair of congruent sides. D. Segment 𝐴𝐵 is a perpendicular bisector of 𝐷𝐶.

260 | Unit 3

5. Draw the image of ∆𝐴𝐵𝐶 after this sequence of rigid transformations. a. Reflect across 𝐴𝐵.

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b. Translate by directed line segment 𝑢.

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6. Triangle 𝐴𝐵𝐶 is congruent to ∆𝐴′𝐵′𝐶′. Describe a sequence of rigid motions that takes 𝐴 to 𝐴′, 𝐵 to 𝐵′, and 𝐶 to 𝐶′.

Unit 3 | 261

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Unit 4: Similarity

Unit 4 | 263

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Unit 4, Lesson 1: Dilations

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Warm-Up: Is That the Same Hippo?

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1. Diego took a picture of a hippo and then edited it. Which is the distorted image? How can you tell?

2. Is there anything about the pictures you could measure to test whether there’s been a distortion?

Unit 4 | 265

Exploration Activity: Sketching and Stretching A dilation with center 𝑂 and positive scale factor 𝑟 takes a point 𝑃 along the ray 𝑂𝑃 to another point whose distance is 𝑟 times farther away from 𝑂 than 𝑃 is. If 𝑟 is less than 1 then the new point is really closer to 𝑂, not farther away.

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1. Dilate 𝐻 using 𝐶 as the center and a scale factor of 3. 𝐻 is 40 millimeters (mm) from 𝐶.

2. Dilate 𝐾 using 𝑂 as the center and a scale factor of 3 . 𝐾 is 40 mm from 𝑂.

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Collaborative Activity: All the Scale Factors

Here is a center of dilation and a triangle.

266 | Unit 4

1. Measure the sides of ∆𝐸𝐹𝐺 (to the nearest mm).

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Your teacher will assign you a scale factor. Predict the relative lengths of the original figure and the image after you dilate by your scale factor.

2. Dilate ∆𝐸𝐹𝐺 using center 𝐶 and your scale factor.

3. How does your prediction compare to the image you drew?

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4. Use tracing paper to copy point 𝐶, ∆𝐸𝐹𝐺, and your dilation. Label your tracing paper with your scale factor.

5. Align your tracing paper with your partner’s. What do you notice?

Unit 4 | 267

Exploration Activity: What Stays the Same?

cr Ratio

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2. Complete the table.

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1. Dilate quadrilateral 𝐴𝐵𝐶𝐷 using center 𝑃 and your scale factor.

Value

𝑃𝐴′ 𝑃𝐴

𝑃𝐵′ 𝑃𝐵

𝑃𝐶′ 𝑃𝐶

𝑃𝐷′ 𝑃𝐷

𝐷′𝐶′ 𝐷𝐶

𝐴′𝐷′ 𝐴𝐷

3. What do you notice? Can you prove your conjecture?

4. Complete the table.

Ratio Value

268 | Unit 4

𝐵′𝐴′ 𝐵𝐴

𝐶′𝐵′ 𝐶𝐵

5. What do you notice? Does the same reasoning you just used also prove this conjecture?

This lesson explored scaled copies.

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Lesson Summary

A scaled copy is a copy of a figure where every length in the original figure is multiplied by the same number.

Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor.

A scale factor is the constant that is multiplied by the length of each side of a figure to produce an image that is the same shape as the original figure.

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A scale factor greater than 1 enlarges an object, and a scale factor less than 1 shrinks an object. A dilation is a transformation that involves scaling figures by a scale factor.

A dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point. The fixed point is the center of the dilation. All of the original distances are multiplied by the same scale factor.

To perform a dilation, a center of dilation, a scale factor, and an object to dilate are needed. The center of a dilation is a fixed point on a plane. It is the starting point from which we measure distances in a dilation.

A dilation with center 𝐴 and positive scale factor 𝑘 takes a point 𝐷 along 𝐴𝐷 to another point whose distance is 𝑘 times farther away from 𝐴 than 𝐷 is. For example, if 𝐹𝐺 is a dilation of 𝐷𝐸 using center 𝐴 and a scale factor of 3, 𝐹𝐴 = 3 ⋅ 𝐷𝐴. If 𝐷𝐴 = 15, then 𝐹𝐴 = 45. © Accelerate Learning Inc. - All Rights Reserved

Unit 4 | 269

Consider another example. A dilation with center 𝑃 and positive scale factor 𝑘 takes a point 𝐴 along 𝑃𝐴 to another point whose distance is 𝑘 times farther away from 𝑃 than 𝐴 is. • Triangle 𝐴′𝐵′𝐶′ is a dilation of ∆𝐴𝐵𝐶 with center 𝑃 and a scale factor of 2, so 𝐴′ is 2 times farther away from 𝑃 than 𝐴 is, 𝐵′ is 2 times farther away from 𝑃 than 𝐵 is, and 𝐶′ is 2 times farther away from 𝑃 than 𝐶 is. Because of the way dilations are defined, all these

quotients give the scale factor: 𝑃𝐴′ = 𝑃𝐵′ = 𝑃𝐶′ = 2. 𝑃𝐵

𝑃𝐶

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𝑃𝐴

• If ∆𝐴𝐵𝐶 is dilated from point 𝑃 with scale factor 1 , then 3 each vertex in ∆𝐴″𝐵″𝐶″ is on the ray from 𝑃 through the corresponding vertex of ∆𝐴𝐵𝐶, and the distance from 𝑃 to each vertex in ∆𝐴″𝐵″𝐶″ is 1 as far as the distance from 𝑃 to 3 the corresponding vertex in ∆𝐴𝐵𝐶.

𝑃𝐴″ = 𝑃𝐵″ = 𝑃𝐶″ = 1 3 𝑃𝐴 𝑃𝐵 𝑃𝐶

The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor. In other words, if 𝐴𝐵 is dilated from point 𝑃 with scale factor 𝑘, then the length of 𝐴𝐵 multiplied by 𝑘 produces the corresponding length of 𝐴′𝐵′.

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𝐴″𝐵″ = 𝐵″𝐶″ = 𝐴″𝐶″ = 𝑘 𝐴𝐵 𝐵𝐶 𝐴𝐶

Corresponding side lengths of the original figure and dilated image are all in the same proportion, related by the same scale factor 𝑘.

Practice Problems

1. Polygon 𝑄 is a scaled copy of Polygon 𝑃.

a. The value of 𝑥 is 6, what is the value of 𝑦?

b. What is the scale factor? 270 | Unit 4

2. Figure 𝑓 is a scaled copy of Figure 𝑒.

We know 𝐴𝐵 = 6, 𝐶𝐷 = 3, 𝑋𝑌 = 4, and 𝑍𝑊 = 𝑎. Select all true equations.

□ 63 = 𝑎4 □ 64 = 𝑎3

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□ 34 = 𝑎6 □ 63 = 𝑎4 □ 64 = 𝑎3 □ 34 = 𝑎6 5

𝑥

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b. 4 = 𝑥

a. 2 = 𝑥

3. Solve each equation.

c. 7 = 28 d. 11 = 5

4. Triangle 𝐴𝐵𝐶 is taken to ∆𝐴′𝐵′𝐶′ by a dilation. Which of these scale factors for the dilation would result in an image that was larger than the original figure? A. 3 5

B. 13 17

C. 1

D. 4 3

Unit 4 | 271

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5. Dilate Figure 𝐺 using center 𝐵 and scale factor 3.

Review Problems

6. Select the shape that has 180° rotational symmetry. A. Rhombus

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D. Quadrilateral

C. Isosceles trapezoid

B. Trapezoid

7. In the figure shown, lines 𝑓 and 𝑔 are parallel. Select all angles that are congruent to angle 1.

□ 1 □ 2 □ 3 □ 4 □ 5 □ 6 □ 7 □ 8

272 | Unit 4

Unit 4, Lesson 2: Dilating Lines and Angles Warm-Up: Angle Articulation Triangle 𝐴′𝐵′𝐶′ is a dilation of ∆𝐴𝐵𝐶 using center 𝑃 and scale factor 2.

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1. What do you think is true about the angles in ∆𝐴′𝐵′𝐶′ compared to the angles in ∆𝐴𝐵𝐶?

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2. Use the tools available to figure out if what you thought was true is definitely true for these triangles.

3. Do you think it would be true for angles in any dilation?

Unit 4 | 273

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Exploration Activity: Dilating Lines

1. Dilate point 𝐴 using center 𝐶 and scale factor 3 . 4

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2. Dilate point 𝐵 using center 𝐶 and scale factor 1 . 3

3. Dilate point 𝐷 using center 𝐶 and scale factor 3 . 4. Dilate 𝐶𝐸 using center 𝐶 and scale factor 2.

5. What happens when the center of dilation is on a line and then you dilate the line?

274 | Unit 4

Collaborative Activity: Proof in Parallel Jada dilated ∆𝐴𝐵𝐶 using center 𝑃 and scale factor 2.

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1. Jada claims that all the segments in ∆𝐴𝐵𝐶 are parallel to the corresponding segments in ∆𝐴′𝐵′𝐶′. Write Jada’s claim as a conjecture.

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2. Prove your conjecture.

3. In Jada’s diagram the scale factor was greater than 1. Would your proof have to change if the scale factor was less than 1?

Unit 4 | 275

Lesson Summary When one figure is a dilation of the other, corresponding side lengths of the original figure and dilated image are in the same proportion and are all related by the same scale factor, 𝑘. What is the relationship of corresponding angles in the original figure and dilated image?

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If ∆𝐴𝐵𝐶 is dilated using center 𝑃 with scale factor 2, it can be verified experimentally that each angle in ∆𝐴𝐵𝐶 is congruent to its corresponding angle in ∆𝐴′𝐵′𝐶′. ∠𝐴 ≅ ∠𝐴′, ∠𝐵 ≅ ∠𝐵′, ∠𝐶 ≅ ∠𝐶′

What is the image of a line not passing through the center of dilation? For example, what will be the image of 𝐵𝐶 when it is dilated with center 𝑃 and scale factor 2?

Congruent corresponding angles can be used to show that 𝐵𝐶 is taken to parallel 𝐵′𝐶 ′.

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What is the image of a line passing through the center of dilation? For example, what will be the image of 𝐺𝐻 when it is dilated with center 𝐶 and scale factor 1 ? 2

When 𝐺𝐻 is dilated with center 𝐶 and scale factor 1 , 𝐺𝐻 is unchanged because, by 2 definition, dilations take points on a line through the center of dilation to points on the same line.

Therefore, a dilation takes a line not passing through the center of the dilation to a parallel line, and it leaves a line passing through the center unchanged.

276 | Unit 4

Practice Problems 1. Angle 𝐴𝐵𝐶 is taken by a dilation with center 𝑃 and scale factor 3 to ∠𝐴′𝐵′𝐶′. The measure of ∠𝐴𝐵𝐶 is 21°. What is the measure of ∠𝐴′𝐵′𝐶′?

□ 𝑙 □ 𝑚 □ 𝑛 □ 𝑜 □ 𝑝

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2. Select all lines that could be the image of line 𝑚 by a dilation.

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A. 𝐴

3. Dilate line 𝑓 with a scale factor of 2. The image is line 𝑔. Which labeled point could be the center of this dilation?

B. 𝐵 C. 𝐶

D. 𝐷

Review Problems 4. Quadrilateral 𝐴′𝐵′𝐶′𝐸′ is the image of quadrilateral 𝐴𝐵𝐶𝐸 after a dilation centered at 𝐹. What is the scale factor of this dilation?

Unit 4 | 277

5. A polygon has a perimeter of 18 units. It is dilated with a scale factor of 3 . What is 2 the perimeter of its image? A. 12 units B. 24 units C. 27 units

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D. 30 units 6. Solve the equation.

4 = 10 7 𝑥

7. Here are some measurements for ∆𝐴𝐵𝐶 and ∆𝑋𝑌𝑍. • The measure of ∠𝐶𝐴𝐵 and ∠𝑍𝑋𝑌 are both 30°.

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• 𝐴𝐶 and 𝑋𝑍 both measure 3 units. • 𝐶𝐵 and 𝑍𝑌 both measure 2 units.

Andre thinks these triangles must be congruent. Clare says she knows they might not be congruent. Draw 2 triangles with the given measurements that aren’t congruent. Explain why triangles with 3 congruent parts aren’t necessarily congruent.

278 | Unit 4

Unit 4, Lesson 3: Splitting Triangle Sides with Dilation – Part 1 Warm-Up: Notice and Wonder: Midpoints

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Here’s ∆𝐴𝐵𝐶 with midpoints 𝐿, 𝑀, and 𝑁.

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What do you notice? What do you wonder?

Unit 4 | 279

Exploration Activity: Dilation or Violation?

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Triangle 𝐴𝐵𝐶 is shown, where points 𝑀 and 𝑁 are the midpoints of 2 sides.

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1. If ∆𝐴𝐵𝐶 is a dilation of ∆𝐴𝑀𝑁, what is the center of dilation and the scale factor?

2. Explain to your partner how ∆𝐴𝐵𝐶 could be a dilation of ∆𝐴𝑀𝑁 with the center and scale factor you found. 3. With your partner, check the definition of dilation, and discuss whether ∆𝐴𝐵𝐶 fits the definition.

4. Explain to your partner how 𝐵𝐶 could be twice as long as 𝑀𝑁.

5. Given that point 𝑀 is the midpoint of 𝐴𝐵 and point 𝑁 is the midpoint of 𝐴𝐶, show that 𝐵𝐶 = 2𝑀𝑁.

280 | Unit 4

Collaborative Activity: A Little Bit Farther Now Here’s ∆𝐴𝐵𝐶. 𝑀 is 2 of the way from 𝐴 to 𝐵. 𝑁 is 2 of the way from 𝐴 to 𝐶. 3

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What can you say about 𝑀𝑁, compared to 𝐵𝐶? Provide a reason for each of your conjectures.

Unit 4 | 281

Lesson Summary Consider a segment whose endpoints are the midpoints of 2 sides of a triangle. If 𝐷 is the midpoint of 𝐵𝐶 and 𝐸 is the midpoint of 𝐵𝐴, then what can be said about 𝐸𝐷 and ∆𝐴𝐵𝐶?

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• 𝐵𝐷 ≅ 𝐷𝐶, and 𝐵𝐸 ≅ 𝐸𝐴.

• Segment 𝐸𝐷 is parallel to the third side of the triangle and half the length of the third side of the triangle. For example, if 𝐴𝐶 = 10, then 𝐸𝐷 = 5. This happens because the entirety of ∆𝐸𝐵𝐷 is a dilation of ∆𝐴𝐵𝐶 with a scale factor of 1 . 2

𝐺𝐴

𝐹𝐶

• In ∆𝐴𝐵𝐶, 𝐹𝐺 divides 𝐴𝐵 and 𝐶𝐵 proportionally. In other words, 𝐵𝐺 = 𝐵𝐹 .

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Again, there is a dilation that takes ∆𝐴𝐵𝐶 to ∆𝐺𝐵𝐹, so 𝐹𝐺 is parallel to 𝐴𝐶, and its length can be calculated using the same scale factor.

Practice Problems

1. What is the measure of angle ∠𝐴′𝐵′𝐶? A. 20° B. 40° C. 60° D. 80°

282 | Unit 4

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2. Triangle 𝐷𝐸𝐹 is formed by connecting the midpoints of the sides of ∆𝐴𝐵𝐶. The lengths of the sides of ∆𝐷𝐸𝐹 are shown. What is the length of 𝐴𝐵?

Review Problems

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3. Angle 𝐴𝐵𝐶 is taken by a dilation with center 𝑃 and scale factor 1 to ∠𝐴′𝐵′𝐶′. The 3 measure of ∠𝐴𝐵𝐶 is 21°. What is the measure of ∠𝐴′𝐵′𝐶′?

4. Draw 2 lines that could be the image of line 𝑚 by a dilation. Label the lines 𝑛 and 𝑝.

Unit 4 | 283

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5. Is it possible for polygon 𝐴𝐵𝐶𝐷𝐸 to be dilated to figure 𝑉𝑊𝑋𝑌𝑍? Explain your reasoning.

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6. Triangle 𝑋𝑌𝑍 is scaled and the image is ∆𝑋′𝑌′𝑍′. Write 2 equations that could be used to solve for 𝑎.

284 | Unit 4

Unit 4, Lesson 4: Connecting Similarity and Transformations

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Warm-Up: Dilation Miscalculation

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What’s wrong with this dilation? Why is 𝐺𝐻𝐹𝐸 not a dilation of 𝐴𝐷𝐶𝐵?

Unit 4 | 285

Exploration Activity: Card Sort: Not-So-Rigid Transformations 1. Your teacher will give you a set of cards. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.

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2. Your teacher will assign you one card. Write the sequence of transformations (translation, rotation, reflection, dilation) to take one figure to the other.

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3. For all the cards that could include a dilation, what scale factor is used to go from Figure 𝐹 to Figure 𝐺? What scale factor is used to go from Figure 𝐺 to Figure 𝐹?

286 | Unit 4

Guided Activity: Similar Triangles? Are the triangles similar?

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𝐴𝐵 ∥ 𝑄𝑅, 𝐴𝐵 ⊥ 𝐴𝐸, 𝑄𝑅 ⊥ 𝑄𝑇

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1. Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other.

2. Write a similarity statement about the 2 figures, and explain how you know they are similar.

3. Compare your statement with your partner’s statement. Is there more than one correct way to write a similarity statement? Is there a wrong way to write a similarity statement?

Unit 4 | 287

Lesson Summary This lesson revisited the concept of similar figures. One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.

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For example, ∆𝐷𝐻𝐹 is similar to ∆𝐸𝐻𝐺. What is a rotation and a dilation that will take ∆𝐷𝐻𝐹 onto ∆𝐸𝐻𝐺? • The triangles are similar because a 180° rotation of ∆𝐷𝐻𝐹 using center 𝐻 will take 𝐻𝐹 onto 𝐻𝐺, since 180° rotations take lines through the center of the rotation to themselves. It will also take 𝐻𝐷 onto 𝐻𝐸 for the same reason. Then, 𝐺 will be on a ray from 𝐻 through 𝐹′, and 𝐸 will be on a ray from 𝐻 through 𝐷′. 𝐻𝐺

𝐻𝐸

𝑃𝐶

• Since 𝐻′𝐹′ = 𝐻′𝐷′ = 𝑃′𝐶′ = 1 , a dilation by a scale factor of 2 will take ∆𝐷′𝐻′𝐹′ onto

∆𝐸𝐻𝐺, which means there is a sequence of rigid motions and dilations that takes ∆𝐷𝐻𝐹 onto ∆𝐸𝐻𝐺.

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• Since similar figures are the result of rigid motions and dilations, all pairs of corresponding angles in similar figures are congruent, and the lengths of all pairs of their corresponding sides are in the same proportion.

• Therefore, ∠𝐷 ≅ ∠𝐸, ∠𝐹 ≅ ∠𝐺, ∠𝐷𝐻𝐹 ≅ ∠𝐸𝐻𝐺, and 𝐻𝐷 = 𝐻𝐹 = 𝐷𝐹 𝐻𝐺

𝐸𝐺

𝐻𝐸

The symbol ∼ is read as “is similar to,” so ∆𝐷𝐻𝐹 ∼ ∆𝐸𝐻𝐺 is read as “triangle 𝐷𝐻𝐹 is similar to triangle 𝐸𝐻𝐺.” Notice that in a similarity statement, corresponding vertices of each triangle must be listed in the same order.

288 | Unit 4

Practice Problems

2. Quadrilaterals 𝑄 and 𝑃 are similar.

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1. Find a sequence of rigid motions and dilations that takes square 𝐴𝐵𝐶𝐷 to square 𝐸𝐹𝐺𝐻.

a. What is the scale factor of the dilation that takes 𝑃 to 𝑄?

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b. What is the scale factor of the dilation that takes 𝑄 to 𝑃?

3. What is our definition of similarity?

A. If 2 figures have the same angles, then they are similar. B. If 2 figures have proportional side lengths, then they are similar. C. If there is a sequence of rigid transformations taking one figure to another, then they are similar. D. If there is a sequence of rigid transformations and dilations that take one figure to the other, then they are similar.

Unit 4 | 289

Review Problems 4. Triangle 𝐷𝐸𝐹 is formed by connecting the midpoints of the sides of ∆𝐴𝐵𝐶. The lengths of the sides of ∆𝐷𝐸𝐹 are shown. What is the length of 𝐵𝐶? B. 4 units C. 6 units D. 8 units

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5. If 𝐴𝐵 is 12, what is the length of 𝐴′𝐵′?

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A. 3 units

6. Right angle 𝐴𝐵𝐶 is taken by a dilation with center 𝑃 and scale factor 1 to ∠𝐴′𝐵′𝐶′. 2 What is the measure of ∠𝐴′𝐵′𝐶′?

290 | Unit 4

7. a. Dilate point 𝐶 using center 𝐷 and scale factor 3 . 4

b. Dilate segment 𝐴𝐵 using center 𝐷 and scale factor 1 .

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A. 1 2

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B. 2

8. A polygon has perimeter 12. It is dilated with a scale factor of 𝑘 and the resulting image has a perimeter of 8. What is the scale factor?

C. 3 4

D. 4

Unit 4 | 291

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Unit 4, Lesson 5: Reasoning about Similarity with Transformations Warm-Up: Notice and Wonder: Nested Triangles

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What do you notice? What do you wonder?

Unit 4 | 293

Exploration Activity: Stretched or Distorted? Triangles

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1. Sketch 2 triangles with all pairs of corresponding angles congruent, and with all pairs of corresponding side lengths in the same proportion.

2. Label your triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 so that ∠𝐴 is congruent to ∠𝐷, ∠𝐵 is congruent to ∠𝐸, and ∠𝐶 is congruent to ∠𝐹. Label each side with its length.

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3. Do the 2 triangles you drew fit the definition of similar? Explain your reasoning.

4. Switch sketches with your partner. Find a sequence of rigid motions and dilations that will take one of their triangles onto the other. Will the same sequence work for your triangles?

294 | Unit 4

Collaborative Activity: Invisible Triangles: Similarity Player 1: You are the transformer. Take the transformer card.

Player 2: Select a triangle card. Do not show it to anyone. Study the diagram to figure out which sides and which angles correspond. Tell Player 1 what you have figured out.

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Player 1: Take notes about what they tell you so that you know which parts of their triangles correspond. Think of a sequence of rigid motions and dilations you could tell your partner to get them to take one of their triangles onto the other. Be specific in your language. The notes on your card can help with this.

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Lesson Summary One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. By the properties of dilations and rigid motions, the following statement is true.

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If figures are similar, then corresponding angles are congruent and pairs of corresponding side lengths are in the same proportion. In the case of triangles, the converse of this statement is also true.

The converse of an if-then statement is the statement that interchanges the hypothesis and the conclusion.

If a pair of triangles has pairs of corresponding side lengths that are all in the same proportion and pairs of corresponding angles that are all congruent, then the triangles must be similar.

Imagine any pair of triangles with all pairs of corresponding side lengths in the same proportion and all pairs of corresponding angles congruent. The same sequence of rigid motions and dilations will work to show that the triangles are similar. For example, ∆𝐼𝐸𝐹 was dilated using 𝐸 as the center by the scale factor given by 𝐵𝐶 . Because the scale factor was chosen this way, it can be

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𝐸𝐹

stated that side 𝐵𝐶 is congruent to side 𝐸′𝐹′.

• Because all pairs of corresponding angles are congruent, there is enough information to use the angle-side-angle (ASA) congruence conditions to prove that ∆𝐼′𝐸′𝐹′ ≅ ∆𝐴𝐵𝐶.

• That means that ∆𝐴𝐵𝐶 can be lined up exactly with a dilation of ∆𝐼𝐸𝐹, which is the definition of similarity.

It doesn’t matter what the triangles look like or where you start. A dilation that makes one pair of corresponding sides congruent can always be defined, and then the ASA congruence conditions can be used to finish proving that there is a sequence of dilations and rigid motions that takes one triangle onto the other. 296 | Unit 4

Practice Problems

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1. Sketch a figure that is similar to this figure. Label side and angle measures.

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2. Write 2 different sequences of transformations that would show that ∆𝐴𝐵𝐶 and ∆𝐴𝐸𝐷 are similar. The length of 𝐴𝐶 is 6 units, 𝐴𝐶 = 6.

Review Problems 3. What is the definition of similarity?

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4. Select all figures which are similar to Parallelogram 𝑃.

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Parallelogram 𝑃

Figure 𝐴

Figure 𝐷

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Figure 𝐶

Figure B

Figure 𝐸

□ Figure 𝐴 □ Figure 𝐵 □ Figure 𝐶 □ Figure 𝐷 □ Figure 𝐸

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5. Find a sequence of rigid transformations and dilations that takes square 𝐴𝐵𝐶𝐷 to square 𝐸𝐹𝐺𝐻.

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A. Translate by the directed line segment 𝐴𝐸, which will take 𝐵 to a point 𝐵′. Then rotate with center 𝐸 by ∠𝐵′𝐸𝐹. Finally, dilate with center 𝐸 by scale factor 5 . 2

B. Translate by the directed line segment 𝐴𝐸, which will take 𝐵 to a point 𝐵′. Then rotate with center 𝐸 by ∠𝐵′𝐸𝐹. Finally, dilate with center 𝐸 by scale factor 2 . 5

C. Dilate using center 𝐸 by scale factor 2 . 5

D. Dilate using center 𝐸 by scale factor 5 .

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6. Triangle 𝐷𝐸𝐹 is formed by connecting the midpoints of the sides of ∆𝐴𝐵𝐶. What is the perimeter of ∆𝐴𝐵𝐶?

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7. Triangles 𝐹𝐴𝐷 and 𝐷𝐶𝐸 are each translations of triangle 𝐴𝐵𝐶.

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Explain why ∠𝐶𝐴𝐷 has the same measure as ∠𝐴𝐶𝐵.

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Unit 4, Lesson 6: Dilations on the Coordinate Plane Warm-Up: Dilating Out

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Dilate triangle 𝐹𝐺𝐻 using center 𝐶 and a scale factor of 3.

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Collaborative Activity: Dilations 1. Quadrilateral 𝐾𝑅𝑀𝑌 was created by dilating 𝑊𝑇𝐷𝐻. The ratio of 𝑊𝑇 to 𝐾𝑅 is 5 .

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a. Use the word bank to complete the statements that follow.

center of dilation

𝑊𝑇𝐷𝐻 is the

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The

scale factor .

𝐾𝑅𝑀𝑌 is the

preimage

Point 𝑃 is the

image

. . is 3 . 5

b. Use the diagram to determine each ratio.

What is the ratio of 𝑊𝐻 ? 𝐾𝑌

What is the ratio of 𝑃𝑅 ? 𝑅𝑇

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Guided Activity: Algebraic Description of a Dilation

a. Complete the table. Ordered Pair

Ordered Pair of the Dilation

Scale Factor

Point

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1. Points 𝐷, 𝐻, 𝐾, and 𝑇 are shown on the coordinate grid. Each point has been dilated using (0, 0) as the center of dilation.

𝐷

𝐻

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𝐾 𝑇

On the coordinate plane, the algebraic representation of a dilation centered at the origin is (𝑥, 𝑦) → (𝑘𝑥, 𝑘𝑦), where 𝑘 is the scale factor.

b. Write the algebraic representation for each point. Point

Algebraic Representation of Dilation

𝐷

𝐻 𝐾 𝑇

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2. Triangle 𝑅′𝐵′𝐾′ is the result of dilating ∆𝑅𝐵𝐾 using the origin as the center of dilation.

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a. What is the scale factor of the dilation?

b. Write the algebraic description of the dilation.

c. Draw a line segment from point 𝑀 that passes through 𝐵′ and ends at 𝐵.

d. The table shows different relationships between line segments 𝑀𝐵, 𝐵′𝐵, and 𝑀𝐵′. Slope

Distance

𝑀𝐵′

3 2

22 + 32 = 13

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Line Segment

𝐵′𝐵 𝑀𝐵

6 4 9 6

62 + 42 = 52 = 2 13

62 + 92 = 117 = 3 13

e. Discuss with your partner how the scale factor could be determined using slope. Summarize your discussion.

f. Discuss with your partner how the scale factor could be determined using distance.

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3. The table shows different relationships between the line segments 𝐵𝑅, 𝐵′𝑅′, 𝐾𝑅 and 𝐾′𝑅′. Line Segment

Slope

Distance

𝐵𝑅

−3

32 +32 = 18 = 3 2

−1

𝐵′𝑅′

3 4.5

32 + 4.52 = 29.25

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𝐾𝑅

12 + 12 = 2

1 1.5

𝐾′𝑅′

12 + 1.52 = 3.25

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Discuss with your partner how the scale factor is also found in the relationships between the slopes of 𝐵𝑅 and 𝐵′𝑅′, and 𝐾𝑅 and 𝐾′𝑅′. Write a summary of the relationships.

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4. Triangle 𝑅′𝐾′𝑊′ is the result of dilating ∆𝑅𝐾𝑊 using point 𝐺(1, 3) as the center of the dilation, as shown.

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b. Complete the table.

a. What do you notice about the location of the center of the dilation?

Line Segment 𝐺𝐾′ 𝐺𝐾

𝐾′𝐾

Slope

4 6

Distance 22 + 32 = 4 + 9 = 13

2 3

c. Determine the scale factor of the dilation.

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Collaborative Activity: Describing Dilations

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1. Triangle 𝐾′𝑀′𝑇′ is the result of dilating ∆𝐾𝑀𝑇 using the origin as the center of dilation. The vertices of ∆𝐾𝑀𝑇 are 𝐾(−8, −6), 𝑀(6, 7), and 𝑇(0, 4). The vertices of ∆𝐾′𝑀′𝑇′ are 𝐾′(−12, −9), 𝑀′(9, 10.5), and 𝑇′(0, 6). Write the algebraic description of the dilation.

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a. Show the dilation on the coordinate grid by drawing dotted lines from the center of dilation through each vertex.

2. Triangles 𝑇𝑊𝐾 and 𝑇′𝑊′𝐾′ are shown on the coordinate grid, where ∆𝑇′𝑊′𝐾′ is a dilation of ∆𝑇𝑊𝐾, and point 𝑃 is the center of dilation.

b. What is the scale factor of the dilation?

c. Write the algebraic description of the dilation.

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3. Triangles 𝑇𝐻𝑀 and 𝑇′𝐻′𝑀′ are shown on the coordinate grid, where ∆𝑇′𝐻′𝑀′ is a dilation of ∆𝑇𝐻𝑀.

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Write a description of the dilation by giving the center of dilation and the scale factor.

Lesson Summary

Algebraic descriptions can be used to describe a dilation on the coordinate plane. To use an algebraic description of a dilation, the center of dilation must be the origin.

Triangles 𝑀𝐾𝑅 and 𝑀′𝐾′𝑅′ are shown, where ∆𝑀′𝐾′𝑅′ is a dilation of ∆𝑀𝐾𝑅 using a scale factor of 1.5 centered at the origin, (0, 0). The algebraic description of the dilation is (𝑥, 𝑦) → (1.5𝑥, 1.5𝑦).

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Find the scale factor between 2 figures using either the slopes or the lengths of line segments.

• The ratio of the vertical change (change in 𝑦-values) in the slope from the point of dilation to an image point to the vertical change (change in 𝑦-values) in the slope from the point of dilation to the corresponding preimage point can be used to find the scale factor.

For example, the vertical change from the origin to 𝑅′ is 9. The vertical change from the origin to 𝑅 is 6. The ratio is 9 = 3 = 1.5. 6

• The ratio of the distance from the point of dilation to an image point to the distance from the point of dilation to the corresponding preimage point can also be used to find the scale factor. For example, the distance from the origin to 𝐾′ is 117 = 3 13. The distance from 3 13 the origin to 𝐾 is 52 = 2 13. The ratio is = 3 = 1.5. 2 2 13

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Practice Problems

Adeline’s Algebraic Description (𝑥, 𝑦) → (2𝑥, 2𝑦)

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1. 𝑊𝑋𝑌𝑍 and 𝑊′𝑋′𝑌′𝑍′ are shown on the coordinate plane, where 𝑊′𝑋′𝑌′𝑍′ is a dilation of 𝑊𝑋𝑌𝑍. Adeline wrote the algebraic description of the dilation centered at the origin, but she made an error. Adeline’s description is shown.

Describe and correct the error Adeline made in her algebraic description.

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2. 𝑁𝐷𝐾𝐺 and 𝑁′𝐷′𝐾′𝐺’ are shown on the coordinate grid, where 𝑁′𝐷′𝐾′𝐺′ is a dilation of 𝑁𝐷𝐾𝐺.

Write a description of the dilation by giving the center of dilation and the scale factor.

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3. Line segment 𝑀′𝑇′ has endpoints 𝑀′(−16, 20) and 𝑇′(4, −8), and is the result of the dilation of 𝑀𝑇 centered at the origin. The endpoints of 𝑀𝑇 are 𝑀(−4, 5) and 𝑇(1, −2). Write the algebraic description of the dilation.

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Review Problems

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4. Triangle 𝐴𝐵𝐶 is dilated. The image is ∆𝐴′𝐵′𝐶′, find the value of 𝑥.

5. 𝑊𝑋𝑌𝑍 is a kite. Angle 𝑊𝑋𝑌 has a measure of 94 degrees and angle 𝑍𝑌𝑋 has a measure of 60 degrees. Find the measure of angle 𝑍𝑊𝑌.

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Unit 4, Lesson 7: Sequences that Include Dilations Warm-Up: Stretched or Distorted? Rectangles

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Are these rectangles similar? Explain how you know.

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1. A sequence of transformations of ∆𝑃𝑇𝑅 is shown on the coordinate grid. Triangle 𝑃𝑇𝑅 is dilated and then translated, where ∆𝑃′𝑇′𝑅′ is the dilation and ∆𝑃′′𝑇′′𝑅′′ is the translation of ∆𝑃′𝑇′𝑅′.

(4, 1)

𝑇′

𝑅′

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𝑅

(2, 2)

𝑃′

(2, 2)

∆𝑷′′𝑻′′𝑹′′

𝑃′′

𝑇

(1, 1)

∆𝑷′𝑻′𝑹′

(4, 4)

(4, 0)

𝑇′′

𝑃

∆𝑷𝑻𝑹

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Guided Activity: Sequence of Transformations

(8, 2)

(6, 2)

𝑅′′

(10, 0)

a. Complete the description.

Triangle 𝑃𝑇𝑅 was dilated using a scale factor of

centered at the origin.

The resulting triangle, ∆𝑃′𝑇′𝑅′, was translated

b. Complete the table.

Transformation Dilation Translation

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to create ∆𝑃′′𝑇′′𝑅′′. Algebraic Description (𝑥, 𝑦) → (

(𝑥, 𝑦) → (𝑥

𝑥,

,𝑦

𝑦)

)

c. Determine the coordinates of each triangle if ∆𝑃𝑇𝑅 is first translated and then dilated. Preimage

Translation

Dilation

∆𝑷𝑻𝑹

∆𝑷′𝑻′𝑹′

∆𝑷′′𝑻′′𝑹′′

𝑇

𝑅

(1, 1)

𝑃′

(4, 1)

𝑅′

(2, 2)

𝑇′

𝑃′′ 𝑇′′

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𝑃

𝑅′′

Some sequences of transformations are commutative, which means the transformations can be applied in any order.

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d. Use the tables to determine if the sequence of transformations applied to ∆𝑃𝑇𝑅 is commutative. Work with your partner to write an explanation.

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(−7, 0)

𝐾’

𝑊’ 𝑃’

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𝑃

(−5, 0)

𝑁

(−8, 2)

𝑲′′𝑾′′𝑷′′𝑵′′

𝑊

(−5, 4)

𝑲′𝑾′𝑷′𝑵′

(5, 4)

𝑁’

𝐾′′

(20, 16)

𝑃′′

(28, 0)

(5, 0)

𝑊′′

(8, 2)

𝑁′′

𝐾

𝑲𝑾𝑷𝑵

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2. 𝐾𝑊𝑃𝑁, 𝐾′𝑊′𝑃′𝑁′, and 𝐾′′𝑊′′𝑃′′𝑁′′ are shown on the coordinate grid.

(7, 0)

(20, 0)

(32, 8)

a. Discuss with your partner the first transformation that occurred. Describe how you determined the first transformation.

b. Determine the scale factor in the second transformation.

c. Complete the description of the transformations. 𝐾𝑊𝑃𝑁 was

𝐾′𝑊′𝑃′𝑁′ was dilated using a scale factor of to create 𝐾′′𝑊′′𝑃′′𝑁′′.

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centered at

to create 𝐾′𝑊′𝑃′𝑁′.

d. Complete the table. Transformation

Algebraic Description (𝑥, 𝑦) → (𝑥, 𝑦) →

𝑃

(−7, 0) (−8, 2)

𝑊’ 𝑃’

𝑁’

𝑲′′𝑾′′𝑷′′𝑵′′

𝐾′′

𝑊′′ 𝑃′′

𝑁′′

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𝑁

(−5, 0)

𝐾’

𝑊

(−5, 4)

𝑲′𝑾′𝑷′𝑵′

𝐾

𝑲𝑾𝑷𝑵

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e. Work with your partner to determine if the sequence of transformations applied to 𝐾𝑊𝑃𝑁 is commutative. Complete the table by first applying the dilation and then the other transformation. Then, write a conclusion based on the completed tables.

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Collaborative Activity: Mapping a Dilation

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1. A sequence of transformations of ∆𝑀𝑊𝐵 is shown on the coordinate grid.

Preimage

Rotation

Dilation

∆𝑴′𝑾′𝑩′

∆𝑴′′𝑾′′𝑩′′

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∆𝑴𝑾𝑩

a. Complete the table of values for each triangle.

𝑀

𝑊

(6, 1)

𝐵

(3, 8)

b. Complete the table.

𝑀′

𝑊′ 𝐵′

(−8, 3) (−1, 6)

Transformation

𝑀′′

𝑊′′ 𝐵′′

(−28, 10.5) (−3.5, 21)

Algebraic Description (𝑥, 𝑦) → (𝑥, 𝑦) →

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Lesson Summary This lesson revisited sequences of transformations to introduce dilations.

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A sequence of transformations is a set of translations, rotations, reflections, and dilations on a figure. The transformations are performed in a given order.

Practice Problems

All sequences can be described using written descriptions. Algebraic descriptions can be used to summarize many sequences of transformations on the coordinate plane. To write an algebraic description of a rotation, the center of rotation must be the origin. To write an algebraic description of a reflection, the line of reflection must be the 𝑥-axis, 𝑦-axis, 𝑦 = 𝑥, or 𝑦 = −𝑥. To write an algebraic description of a dilation, the center of dilation must be the origin.

1. Triangles 𝐶𝐾𝑅, 𝐶′𝐾′𝑅′, and 𝐶′′𝐾′′𝑅′′ are shown on the coordinate plane.

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a. Describe the sequence of transformations of ∆𝐶𝐾𝑅 that results in ∆𝐶′′𝐾′′𝑅′′.

b. Write the algebraic description of each transformation.

Transformation

Algebraic Description (𝑥, 𝑦) → (𝑥, 𝑦) →

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2. 𝐿𝐹𝑁𝐶, 𝐿′𝐹′𝑁′𝐶′, 𝐿′′𝐹′′𝑁′′𝐶′′, and 𝐿′′′𝐹′′′𝑁′′′𝐶′′′ are shown on the coordinate grid.

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a. Describe the sequence of transformations of 𝐿𝐹𝑁𝐶 that resulted in 𝐿′′′𝐹′′′𝑁′′′𝐶′′′.

b. Write the algebraic descriptions of the sequence of transformations.

Transformation

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Algebraic Description

(𝑥, 𝑦) → (𝑥, 𝑦) → (𝑥, 𝑦) →

Review Problems 3. Triangles 𝐴𝐵𝐶 and 𝐴𝐸𝐷 are shown, where 𝐴𝐶 = 6. Select all sequences of transformations that would show that ∆𝐴𝐵𝐶 and ∆𝐴𝐸𝐷 are similar.

□ Dilate ∆𝐴𝐵𝐶 using center 𝐴 by a scale factor of 12 , then reflect over 𝐴𝐶

□ Dilate ∆𝐴𝐸𝐷 using center 𝐴 by a scale factor of 2, then reflect over 𝐴𝐶 . scale factor of 1 . 2

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□ Reflect ∆𝐴𝐵𝐶 over 𝐴𝐶 , then dilate using center 𝐴 by a

□ Reflect ∆𝐴𝐸𝐷 over 𝐴𝐶 , then dilate using center 𝐴 by a scale factor of 2.

□ Translate ∆𝐴𝐸𝐷 by directed 𝐷𝐶, then dilate using center 𝐶 by scale factor 2.

□ Translate either ∆𝐴𝐵𝐶 or ∆𝐴𝐸𝐷 by directed line seg-

ment 𝐷𝐶, then reflect over 𝐴𝐶.

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4. Select all true statements given that angle ∠𝐴𝐸𝐷 ≅ ∠𝐴𝐵𝐶.

□ 𝑚∠𝐴𝐶𝐵 = (180 − 𝑥)° □ 𝑚∠𝐴𝐶𝐵 = 𝑥° □ ∆𝐴𝐶𝐵 ~ ∆𝐴𝐷𝐸 □ 𝐴𝐷 = 13 𝐴𝐶 □ 𝐴𝐷 = 12 𝐷𝐶

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Unit 4, Lesson 8: Conditions for Triangle Similarity Warm-Up: Math Talk: Angle-Side-Angle as a Helpful Tool

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1. How could you justify each statement?

a. Triangle 𝑃′𝑄′𝑅′ is congruent to ∆𝑆𝑇𝑈.

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b. Triangle 𝑃𝑄𝑅 is similar to ∆𝑆𝑇𝑈.

c. Triangle 𝐺′𝐻′𝐼′ is congruent to ∆𝑀𝑁𝑂.

d. Triangle 𝐺𝐻𝐼 is similar to ∆𝑀𝑁𝑂.

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Exploration Activity: How Many Pieces? For each problem, draw 2 triangles that have the listed properties. Try to make them as different as possible.

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1. One angle is 45°.

2. One angle is 45° and another angle is 30°.

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3. One angle is 45° and another angle is 30°. The lengths of a pair of corresponding sides are 2 centimeters (cm) and 6 cm.

4. Compare your triangles with your neighbors’ triangles. Which ones seem to be similar no matter what?

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5. Prove your conjecture.

Collaborative Activity: Any Two Angles?

Here are 2 triangles. One triangle has a 60° angle and a 40° angle. The other triangle has a 40° angle and an 80° angle.

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1. Explain how you know the triangles are similar.

2. How long are the sides labeled 𝑥 and 𝑦?

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Lesson Summary When 2 angles of 1 triangle are congruent to 2 angles of another triangle, the 2 triangles are similar. This is called the angle-angle (AA) triangle similarity theorem.

Dilate ∆𝐴𝐵𝐶 by the ratio 𝐷𝐸 so that 𝐴′𝐵′ is 𝐴𝐵

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Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are shown, where ∠𝐴 ≅ ∠𝐷 and ∠𝐵 ≅ ∠𝐸. If a sequence of rigid motions and dilations moves the first figure so that it fits exactly over the second, then the AA triangle similarity has been shown to be true.

congruent to 𝐷𝐸. Now, ∆𝐴′𝐵′𝐶′ ≅ ∆𝐷𝐸𝐹 by the ASA triangle congruence conditions, which means there is a sequence of rotations, reflections, and translations that takes ∆𝐴′𝐵′𝐶′ onto ∆𝐷𝐸𝐹.

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Therefore, a dilation followed by a sequence of rotations, reflections, and translations will take ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹, which is the definition of similarity. It has shown that a dilation and a sequence of rigid motions takes ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹, so the triangles are similar.

Practice Problems

1. What is the length of segment 𝐷𝐹?

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2. In ∆𝐴𝐵𝐶, angle 𝑚∠𝐴 = 35° and 𝑚∠𝐵 = 20°. Select all triangles which are similar to ∆𝐴𝐵𝐶.

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□ ∆𝐷𝐸𝐹, where 𝑚∠𝐷 = 35° and 𝑚∠𝐸 = 20° □ ∆𝐺𝐻𝐼, where 𝑚∠𝐺 = 35° and 𝑚∠𝐼 = 30° □ ∆𝐽𝐾𝐿, where 𝑚∠𝐽 = 35° and 𝑚∠𝐿 = 125° □ ∆𝑀𝑁𝑂, where 𝑚∠𝑁 = 20° and 𝑚∠𝑂 = 125° □ ∆𝑃𝑄𝑅, where 𝑚∠𝑄 = 20° and 𝑚∠𝑅 = 30°

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3. Decide whether triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐶 are similar. Explain or show your reasoning.

Review Problem

4. Quadrilaterals 𝑄 and 𝑃 are similar.

What is the scale factor of the dilation that takes 𝑃 to 𝑄? A. 3

B. 4 5

C. 5 4

D. 5 3

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Unit 4, Lesson 9: Other Conditions for Triangle Similarity Warm-Up: Math Talk: Triangle Congruence 1. For each figure, evaluate the given information mentally while considering the questions listed. • Is there enough information to determine if the pairs of triangles are congruent?

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• If so, which theorem(s) would you use?

• If not, what additional piece of information could you use?

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b. 𝐻𝐼 ≅ 𝐹𝐺

a. 𝐾𝑀 ⊥ 𝑁𝐿 and 𝐾𝐿 ≅ 𝑀𝐿

c. 𝐴𝐵 ∥ 𝐶𝐷 and ∠𝐷𝐴𝐶 ≅ ∠𝐵𝐶𝐴

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Exploration Activity: Side-Angle-Side Triangle Similarity? Andre remembers lots of ways to prove triangles congruent. He asks Clare, “Can we use Angle-Side- Angle to prove triangles are similar?” Clare: “Sure, but we don’t need the Side part because Angle-Angle is enough to prove triangles are similar.”

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Andre: “Hmm, what about Side-Angle-Side or Side-Side-Side? What if we don’t know 2 angles?” Clare: “Oh! I don’t know. Let’s draw a picture and see if we can prove it.” Andre: “Uh-oh. If ‘side’ means corresponding sides with the same length, then we’ll only get congruent triangles.”

1. What could ‘side’ stand for to prove triangles similar?

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2. Draw a diagram that would help you prove the Side-Angle-Side Triangle Similarity Theorem.

3. Write a proof.

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Guided Activity: Side-Side-Side Triangle Similarity Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are shown. 1. Complete the proof.

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Given: 𝐴𝐵 = 𝑐, 𝐴𝐶 = 𝑏, 𝐵𝐶 = 𝑎, 𝐷𝐸 = 𝑘𝑐, 𝐷𝐹 = 𝑘𝑏, and 𝐸𝐹 = 𝑘𝑎. Prove: ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹

Dilate ∆𝐴𝐵𝐶 by scale factor 𝑘 centered at point 𝐴 to create ∆𝐴′𝐵′𝐶′. Since all 3 pairs of corresponding sides are scaled by scale factor 𝑘, 𝐴′𝐵′ = 𝐷𝐸, 𝐵′𝐶′ = . Therefore, ∆𝐴′𝐵′𝐶′ ≅ ∆𝐷𝐸𝐹 by

𝐴′𝐶′ =

there is a sequence of

∆𝐷𝐸𝐹. This means ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹 because there is a

, and

, so

that takes ∆𝐴′𝐵′𝐶′ onto

and a

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sequence of rigid motions that takes ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹. 2. Complete the statement.

Any pair of triangles with

pairs of corresponding proportional

must be similar.

3. What do you know about the value of 𝑘 in this proof?

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Lesson Summary Besides the angle-angle (AA) triangle similarity explored in the previous lesson, what other conditions are sufficient to prove triangles similar?

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• When 2 sides of 1 triangle are proportional to 2 corresponding sides of another triangle using the same scale factor 𝑘, and the pair of angles between these sides are congruent, the triangles are similar by the side-angle-side (SAS) triangle similarity theorem.

For example, in the triangles shown, ∠𝐸𝐷𝐹 and ∠𝐵𝐷𝐶 are vertical angles and therefore congruent, and there are 2 pairs of corresponding sides with the same scale factor.

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Dilate ∆𝐷𝐸𝐹 using center 𝐷 and scale factor 𝑘. Since 𝐵𝐷 = 𝐶𝐷 = 𝑘, 𝐵𝐷 is now 𝐸𝐷

𝐹𝐷

congruent to 𝐸′𝐷, and 𝐶𝐷 is congruent to 𝐹′𝐷. Since the dilation did not change the size of the angles, ∆𝐸′𝐷𝐹′ ≅ ∆𝐵𝐷𝐶 by the SAS triangle congruence conditions. This means there is a sequence of rigid motions that takes ∆𝐸′𝐷𝐹′ onto ∆𝐵𝐷𝐶. That means ∆𝐵𝐷𝐶 ~ ∆𝐸𝐷𝐹 because there is a dilation and a sequence of rigid motions that takes one onto the other. No additional information was given about these triangles. Therefore, any pair of triangles that have 2 pairs of sides whose lengths are in the same proportion and have a congruent angle between them must be similar.

• It can also be shown that if all 3 pairs of corresponding sides are proportional and use the same scale factor 𝑘, this is sufficient to prove the triangles are similar. This is called the side-side-side (SSS) triangle similarity theorem.

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Practice Problems 1. Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are shown, with side lengths labeled.

b. What does that tell us about ∠𝐷?

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a. Explain how we know that ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹 are similar.

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2. Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are shown, with measurements labeled.

a. Find the length of 𝐸𝐹.

b. Find the measure of ∠𝐸.

c. Find the measure of ∠𝐹.

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Review Problems

B. 81 units 4

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C. 36 units

A. 3 units

4. What is the length of segment 𝐷𝐹?

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3. Decide whether triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐶 are similar. Explain or show your reasoning.

D. 48 units

5. In ∆𝐴𝐵𝐶, 𝑚∠𝐴 = 75° and 𝑚∠𝐵 = 20°. Which triangle is similar to ∆𝐴𝐵𝐶? A. ∆𝐷𝐸𝐹, where 𝑚∠𝐷 = 75° and 𝑚∠𝐸 = 20°

B. ∆𝐷𝐸𝐹, where angle 𝑚∠𝐷 = 20° and 𝑚∠𝐸 = 75° C. ∆𝐷𝐸𝐹, where 𝑚∠𝐷 = 85° and 𝑚∠𝐸 = 20° D. ∆𝐷𝐸𝐹, where 𝑚∠𝐷 = 20° and 𝑚∠𝐹 = 85°

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6. Sketch a pair of rectangles that are similar.

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b. Two angles are similar.

a. Two line segments are similar.

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7. Determine if each statement must be true, could possibly be true, or definitely can’t be true. Explain or show your reasoning.

8. Figure 𝐺′ is the image of Figure 𝐺 by a dilation.

a. Where is the center of this dilation?

b. Estimate the scale factor.

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Unit 4, Lesson 10: Justifying Similarity Warm-Up: Solar Eclipse

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The diameter of the Sun is 1,391,000 kilometers (km). The diameter of the Moon is 3,475 km. The distance from Earth to the Sun is 149,600,000 km.

(not to scale)

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How far would the Moon have to be from Earth for the Moon to appear the same size as the Sun?

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Guided Activity: Similarity in Terms of Non-Rigid Motions 1. Triangles 𝑃𝑊𝐶 and 𝑅𝑇𝐺 are shown.

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a. What do you notice about the two triangles?

b. What transformations can be applied so that ∆𝑊𝑃𝐶 is mapped onto ∆𝑇𝑅𝐺?

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c. What can you conclude from the transformations?

Omario and Emelia were asked, “If ∆𝑃𝑊𝐶 and ∆𝑅𝑇𝐺 are similar by angle-angle similarity, what information would need to be given?” Their answers are shown. Emelia

∠𝑊𝑃𝐶 ≅ ∠𝑇𝑅𝐺 and ∠𝑊𝐶𝑃 ≅ ∠𝑇𝐺𝑅

∠𝑃𝑊𝐶 ≅ ∠𝑅𝑇𝐺 and ∠𝑃𝐶𝑊 ≅ ∠𝑅𝐺𝑇

Omario

d. What do you notice about their answers?

e. Explain why both students answered the question correctly.

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2. Triangles 𝑀𝑅𝐶 and 𝑌𝐻𝐶 are shown.

a. Complete the table to indicate whether each set of given information makes the statement true or false.

Given Information ∠𝐻𝐶𝑌 ≅ ∠𝑅𝐶𝑀

True or False? ∆𝑴𝑹𝑪 ~ ∆𝒀𝑯𝑪 by AA Similarity True

False

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∠𝐻 ≅ ∠𝑅

∠𝐻𝐶𝑌 ≅ ∠𝑅𝐶𝑀

True

∠𝐻 ≅ ∠𝑅

True

𝑀𝑅 = 𝑀𝐶 𝑌𝐻 𝑌𝐶

False

∠𝑌 ≅ ∠𝑀

b. Select all the transformations in a sequence that could be applied so that ∆𝑀𝑅𝐶 coincides with ∆𝑌𝐻𝐶.

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□ rotate ∆𝑀𝑅𝐶 about point 𝐶 □ rotate ∆𝑀𝑅𝐶 about point 𝑅 □ reflect ∆𝑀𝑅𝐶 across 𝑀𝑅 □ reflect ∆𝑀𝑅𝐶 across 𝑌𝐻 □ translate ∆𝑀𝑅𝐶 left and up □ dilate ∆𝑀𝑅𝐶 using point 𝐶 as the center □ dilate ∆𝑀𝑅𝐶 using point 𝑅 as the center

c. Discuss with your partner what is true about the triangles when all the transformations have been applied. d. Use the definition of similarity in terms of non-rigid motions found in the Lesson Summary to justify that ∆𝑀𝑅𝐶 and ∆𝑌𝐻𝐶 are similar.

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Collaborative Activity: Justifying Similarity 1. Consider the following statement.

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If it is possible to find one or more transformations that will transform a figure such that all corresponding points coincide with the other figure, then the figures will be similar. Explain what this statement means in your own words.

2. Triangles 𝑅𝐾𝑊 and 𝐵𝑍𝑆 are shown. Assume 𝑅𝑊 = 𝑊𝐾 = 𝐾𝑅 . 𝐵𝑆

𝑆𝑍

𝑍𝐵

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a. Explain how you can use tracing paper to verify ∆𝑅𝐾𝑊 ~ ∆𝐵𝑍𝑆.

b. Write a sequence of transformations that will map ∆𝑅𝐾𝑊 onto ∆𝐵𝑍𝑆 so that ∆𝑅𝐾𝑊 ~ ∆𝐵𝑍𝑆.

c. If it is given that ∆𝐵𝑍𝑆 is a result of (𝑥, 𝑦) → (−𝑦, 𝑥) and then (𝑥, 𝑦) → (2𝑥, 2𝑦) to ∆𝑅𝐾𝑊. Are the triangles congruent? Are the triangles similar?

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d. Discuss with your partner what criteria would have to be given so that SAS similarity would be sufficient to justify that the triangles are similar. Summarize your discussion.

3. Katelyn, Diego, and Amelia used ∆𝐾𝑅𝑉 and ∆𝑍𝑅𝐴 to draw conclusions about proving triangles similar. Their conclusions are shown.

Diego’s Conclusion

Show that two triangles are congruent using rigid motions and dilation. Then, apply the definition of similarity in terms of non-rigid transformations. This is sufficient to prove the triangles are similar.

Assume ∠𝑉𝐾𝑅 ≅ ∠𝐴𝑍𝑅 and ∠𝑉𝑅𝐾 ≅ ∠𝐴𝑅𝑍.

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Katelyn’s Conclusion

To prove similarity of two triangles, apply the definition of congruence in terms of rigid motions to show that two angles that are assumed to be congruent are congruent. This is sufficient to prove similarity of two triangles.

Amelia’s Conclusion

By applying a series of rigid transformations and a dilation, it could be shown that the triangles are similar by showing that one set of sides is proportional and assuming that two angles are congruent.

a. Whose conclusion do you agree with?

b. Explain why you agree with that student.

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c. Pick one student whose conclusion is incorrect. Write a note to them to help them understand their error.

Lesson Summary

A similarity transformation, or similarity in terms of non-rigid motions, includes a dilation or one or more rigid transformations (reflection, rotation, and/or translation) combined with a dilation that leads to 2 figures being similar.

• Similar figures have the exact same shape but not necessarily the same size or the same orientation.

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• Corresponding angles in similar shapes are congruent, whereas corresponding sides are proportional.

Triangles 𝐵𝑅𝑁 and 𝑀𝐾𝑆 are shown, where ∠𝑁𝐵𝑅 ≅ ∠𝐾𝑆𝑀 and ∠𝑅𝑁𝐵 ≅ ∠𝑀𝐾𝑆. A rotation of 90° counterclockwise about the origin and a dilation with a scale factor of 1 2 centered at the origin can map ∆𝑅𝑁𝐵 onto ∆𝑀𝐾𝑆. Because of similarity in terms of non-rigid motions, ∆𝑅𝑁𝐵 ~ ∆𝑀𝐾𝑆. Since ∠𝑁𝐵𝑅 ≅ ∠𝐾𝑆𝑀 and ∠𝑅𝑁𝐵 ≅ ∠𝑀𝐾𝑆, AA similarity is sufficient to show that the triangles are similar.

340 | Unit 4

Practice Problems 1. Triangle 𝐴𝐵𝐶 is shown.

A sequence of 3 transformations is shown. • (𝑥, 𝑦) → (𝑥 − 4, 𝑦 − 1)

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• (𝑥, 𝑦) → (𝑦, 𝑥)

• (𝑥, 𝑦) → (1.5𝑥, 1.5𝑦)

a. Draw the result of each transformation.

Relationship of Preimage to Image

Transformation

b. Complete the table by noting if the preimage and image are congruent or similar.

(𝑥, 𝑦) → (𝑥 − 4, 𝑦 − 1)

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(𝑥, 𝑦) → (𝑦, 𝑥)

(𝑥, 𝑦) → (1.5𝑥, 1.5𝑦)

2. Select all the transformations that will result in similar figures.

□ (𝑥, 𝑦) → (3𝑥, 3𝑦) □ (𝑥, 𝑦) → (𝑥 + 3, 𝑦 + 3) □ (𝑥, 𝑦) → (𝑦, 𝑥) □ (𝑥, 𝑦) → (−𝑦, 𝑥) □ (𝑥, 𝑦) → � 14 𝑥, 14 𝑦� □ (𝑥, 𝑦) → (1.2𝑥, 1.2𝑦)

Unit 4 | 341

3. Triangle 𝐻𝑌𝑊 is a result of (𝑥, 𝑦) → (−𝑥, −𝑦) and then (𝑥, 𝑦) → � 2 𝑥, 2 𝑦� to ∆𝑍𝑆𝑃. 3 3 Explain whether the triangles are congruent or similar.

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Review Problems

4. Find a sequence of rigid transformations and dilations that takes square 𝐸𝐹𝐺𝐻 to square 𝐴𝐵𝐶𝐷.

5. Polygon 𝑄 is a scaled copy of Polygon 𝑃.

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The value of 𝑥 is 6, what is the value of 𝑦? A. 7 2

B. 4

C. 9 2

D. 5

6. Solve each equation. a. 2 = 𝑥 5

b. 2 = 𝑥 3

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Unit 4, Lesson 11: Splitting Triangle Sides with Dilation – Part 2 Warm-Up: Notice and Wonder: Parallel Segments What do you notice? What do you wonder?

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𝐴𝐶 ∥ 𝑀𝑁

Unit 4 | 343

Exploration Activity: Prove It: Parallel Segments Does a line parallel to one side of a triangle always create similar triangles?

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1. Create several examples. Decide if the conjecture is true or false. If it’s false, make a more specific true conjecture.

2. Find any additional information you can be sure is true.

Label it on the diagram.

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𝐴𝐶 ∥ 𝑀𝑁

3. Write an argument that would show that your conjecture is true.

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Collaborative Activity: Preponderance of Proportional Relationships 1. Find the length of each unlabeled side.

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a. Triangle 𝐷𝐸𝐹 is shown, where 𝐴𝐵 ∥ 𝐸𝐹 and 𝐴𝐷 ⊥ 𝐷𝐵.

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b. Triangle 𝐸𝐹𝐺 is shown, where 𝐵𝐷 ∥ 𝐹𝐺, 𝐸𝐹 = 12, and 𝐸𝐵 = 2.5.

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Lesson Summary In ∆𝐴𝐵𝐶, 𝐹𝐺 ∥ 𝐴𝐶, as shown. Using relationships created when parallel lines are cut by a transversal, it can be shown that the corresponding angles in ∆𝐴𝐶𝐵 and ∆𝐺𝐹𝐵 are congruent. Therefore, the triangles are similar by the AA triangle similarity conditions.

𝐺𝐴

𝐹𝐶

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There must be a dilation that maps ∆𝐺𝐹𝐵 to ∆𝐴𝐶𝐵, and so pairs of corresponding side lengths are in the same proportion. For this reason, it can be shown that 𝐺𝐹 divides 𝐴𝐵 and 𝐶𝐵 proportionally. In other words, 𝐵𝐺 = 𝐵𝐹 .

For example, suppose 𝐺 is 2 of the way from 𝐴 to 𝐵 and 𝐹 is 2 of the way from 𝐶 to 𝐵. 3

Then, if 𝐵𝐴 = 9 and 𝐵𝐶 = 12, it can be determined that 𝐺𝐴 = 6 and 𝐹𝐶 = 8. 𝐺𝐴

𝐹𝐶

What will 𝐵𝐺 and 𝐵𝐹 equal? Since 𝐵𝐺 = 3 and 𝐵𝐹 = 4, it can be shown that 3 = 4 , which 6 8 demonstrates that 𝐵𝐺 = 𝐵𝐹 .

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This argument holds in general. A segment in a triangle that is parallel to 1 side of the triangle divides the other 2 sides of the triangle proportionally.

Practice Problems

1. Segment 𝐴′𝐵′ is parallel to 𝐴𝐵.

a. What is the length of 𝐴𝐵?

b. What is the length of 𝐵′𝐵?

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2. Explain how you know that 𝐷𝐸 is not parallel to 𝐵𝐶.

3. In right triangle 𝐴𝐵𝐶, 𝐴𝐶 = 4 and 𝐵𝐶 = 5. A new ∆𝐷𝐸𝐶 is formed by connecting the midpoints of 𝐴𝐶 and 𝐵𝐶. a. What is the area of ∆𝐴𝐵𝐶?

b. What is the area of ∆𝐷𝐸𝐶?

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c. Does the scale factor for the side lengths apply to the area as well?

Review Problems

4. Which of these statements is true?

A. To know whether 2 triangles are similar, it is enough to know the measure of 1 angle. B. To know whether 2 triangles are similar, it is enough to know the length of 1 side. C. To know whether 2 triangles are similar, it is enough to know the measure of 2 angles in each triangle. D. To know whether 2 triangles are similar, it is enough to know the measure of 2 sides in each triangle. © Accelerate Learning Inc. - All Rights Reserved

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5. Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are shown.

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a. Are ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹 similar? Show or explain your reasoning.

b. If possible, find the length of 𝐸𝐹. If not, explain why the length of 𝐸𝐹 cannot be determined.

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6. What is the length of 𝐷𝐹? ?

7. The ∆𝐴𝐵𝐶 is taken to ∆𝐴′𝐵′𝐶′ by a dilation. Select all of the scale factors for the dilation that would result in an image that was smaller than the original figure.

□ 12 □ 89 □ 1 □ 32 □ 2

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Unit 4, Lesson 12: Proving Triangle Similarity – Part 1 Warm-Up: Describing a Sequence

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1. Triangles 𝑀𝑃𝐶 and 𝑇𝑆𝐿 are shown on the coordinate plane, where ∆𝑀𝑃𝐶 is the preimage.

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Describe a sequence of 2 transformations that will map the triangles.

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Collaborative Activity: Proving Triangle Similarity Work with your partner to complete the following. 1. Triangles 𝑊𝑍𝑌 and 𝑅𝑃𝑇 are shown. Angle 𝑊𝑍𝑌 is congruent to ∠𝑇𝑅𝑃, and ∠𝑍𝑊𝑌 ≅ ∠𝑅𝑇𝑃.

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a. Mark the figure using the given information.

b. Which similarity theorem can be applied to show that the triangles are similar?

d. Complete each statement.

𝑍𝑌 =

𝑃𝑇

𝑅𝑇

∠𝑊𝑌𝑍 ≅

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𝑊𝑍 =

c. Write the similarity statement.

2. Triangles 𝐾𝐻𝐺 and 𝑀𝑉𝐶 are shown. Angle 𝐾𝐻𝐺 is congruent to ∠𝑉𝑀𝐶, and 𝐾𝐻 = 𝐺𝐻 . 𝑉𝑀

𝐶𝑀

a. Mark the figure using the given information.

b. Which similarity theorem can be applied to show that the triangles are similar?

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c. Write the similarity statement.

d. Complete each statement. ∠𝐻𝐺𝐾 ≅

𝐶𝑀

∠𝐶𝑉𝑀 ≅

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𝐾𝐺 =

Guided Activity: Proving Triangle Similarity 1. Triangles 𝐺𝑇𝐴 and 𝑇𝐸𝑅 are shown, and 𝐺𝐴 ∥ 𝐸𝑅.

a. With your partner, discuss what conclusions can be drawn using the given information. Summarize your discussion, and explain how you reached each conclusion.

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b. Which similarity postulates should be used to prove ∆𝐺𝑇𝐴 ~ ∆𝑅𝑇𝐸? c. Complete the two-column proof. Statement

1. 𝐺𝐴 ∥ 𝐸𝑅

Reason

2. If two parallel lines are intersected by a transversal, then alternate interior angles are congruent.

4. ∆𝐺𝑇𝐴 ~ ∆𝑅𝑇𝐸

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d. Ask two classmates about their proofs. Record each person’s step 3 statement, and summarize their reason. You are the only person who should write in the first two columns of the table. Have each person initial next to your summary, indicating that your summary was correct. Step 3 Reason

Initials

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Step 3 Statement

e. Discuss with your partner the two different ways that ∆𝐺𝑇𝐴 ~ ∆𝑅𝑇𝐸 can be proved.

f. Explain why you do not need to state three congruent angles in your two-column proof.

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g. Arian and Isabella were discussing why there is no angle-angle-angle similarity theorem. Arian

I think, once you know two pairs of angles in a triangle are congruent, then you know the last pair is congruent, since the sum of all of the angles of a triangle is 180°.

Isabella I think once you know two pairs of angles in a triangle are congruent, then you know all of the sides are proportional.

Discuss with your partner why Arian and Isabella’s arguments could also be applied to show why angle-side-angle similarity and angle-angle-side similarity are not needed to prove similarity. Summarize your discussion.

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2. Triangles 𝑆𝑃𝐷 and 𝑀𝑃𝑅 are shown.

Given: 𝑀𝑆 = 3, 𝑆𝑃 = 12, 𝑅𝐷 = 2, and 𝐷𝑃 = 8 Prove: ∆𝑆𝐷𝑃 ~ ∆𝑀𝑅𝑃

Complete the two-column proof. Reason

1. 𝑀𝑆 = 3, 𝑆𝑃 = 12, 𝑅𝐷 = 2, and 𝐷𝑃 = 8 2. 𝑀𝑆 + 𝑆𝑃 = 𝑃𝑀, 𝑅𝐷 + 𝐷𝑃 = 𝑅𝑃 4. 𝑃𝑆 = 12, 𝑃𝐷 = 8 15

𝑃𝑀

𝑃𝑅

5. 𝑃𝑆 = 𝑃𝐷

𝑃𝑅

5. 6.

6. ∠𝑃 ≅ ∠𝑃

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7. ∆𝑆𝐷𝑃 ~ ∆𝑀𝑅𝑃

3. Substitution property of equality

3. 𝑃𝑀 = 15, 𝑅𝑃 = 10 𝑃𝑀

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Statement

Lesson Summary

To prove triangles are similar, first consider the diagram and any given information. Think about whether it is easier to find pairs of corresponding angles that are congruent, pairs of corresponding sides that are proportional, or a combination of corresponding sides and angles. Then, check if there’s enough information to use AA similarity, SAS similarity, or SSS similarity conditions. • If 2 angles of a triangle are congruent to 2 angles of another triangle, then the triangles are similar by angle-angle (AA) similarity. • If the lengths of all the corresponding sides of 2 triangles are proportional, then the triangles are similar by side-side-side (SSS) similarity. • If the lengths of 2 sides of each triangle are proportional and their included angles are congruent, then the triangles are similar by side-angle-side (SAS) similarity.

Unit 4 | 353

Practice Problems 1. Triangles 𝑇𝐿𝐺 and 𝑇𝑆𝐷 are shown.

Prove: ∆𝑇𝐿𝐺 ~ ∆𝑇𝑆𝐷

Complete the two-column proof. Statement

Reason

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3. 𝑇𝐷 = 6, 𝑇𝑆 = 9

4. 𝑇𝐺 = 4 , 𝑇𝐿 = 6 , 𝐺𝐿 = 6 6

𝑇𝐷

𝑇𝑆

5. 𝑇𝐺 = 𝑇𝐿 = 𝐺𝐿

𝐷𝑆

3. Substitution property of equality 4. 5. 6.

6. ∆𝑇𝐿𝐺 ~ ∆𝑇𝑆𝐷

𝐷𝑆

2. 𝑇𝐺 + 𝐺𝐷 = 𝑇𝐷, 𝑇𝐿 + 𝐿𝑆 = 𝑇𝑆

1. 𝑇𝐺 = 4, 𝐺𝐷 = 2, 𝑇𝐿 = 6, 𝐿𝑆 = 3, 𝐺𝐿 = 6, and 𝐷𝑆 = 9

𝑇𝐷

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Given: 𝑇𝐺 = 4, 𝐺𝐷 = 2, 𝑇𝐿 = 6, 𝐿𝑆 = 3, 𝐺𝐿 = 6, and 𝐷𝑆 = 9

2. Mai thinks knowing the measures of 2 sides is enough to show triangle similarity. Do you agree? Explain or show your reasoning.

354 | Unit 4

3. Amanda and Tyler are working to determine if ∆𝑀𝐶𝐷 ~ ∆𝑂𝐿𝑁 is true. Their reasonings are shown. Amanda’s Reasoning

Tyler’s Reasoning

No, the triangles are not similar because the corresponding sides are not proportional.

Yes, the triangles are similar because the corresponding sides are proportional.

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Who do you agree with? Explain your reasoning.

Review Problems

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4. Triangle 𝐷𝐸𝐹 is formed by connecting the midpoints of the sides of triangle 𝐴𝐵𝐶. Select all true statements.

□ ∆𝐵𝐷𝐸 ≅ ∆𝐸𝐹𝐶 □ ∆𝐵𝐷𝐸 ≅ ∆𝐷𝐴𝐹 □ 𝐵𝐷 ≅ 𝐹𝐸 □ 𝐵𝐶 = 8 □ 𝐵𝐶 = 6

Unit 4 | 355

5. Which pair of polygons is similar?

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356 | Unit 4

Unit 4, Lesson 13: Proving Triangle Similarity – Part 2 Warm-Up: Would You Rather? 1. In 2021, a major car manufacturer introduced an air taxi and a self-driving car. Both are concepts with no known timeline for production.

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a. Do you think a drone-like personal vehicle like an air taxi will be a reality in your lifetime?

b. Would you rather have an air taxi or a self-driving car in the future?

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c. Explain your choice.

Unit 4 | 357

Collaborative Activity: Proving Triangle Similarity With your partner, sort through the statement and reason cards. Then, write your proofs in the tables provided.

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Proof A

Given: 𝐺𝑆 ∥ 𝑅𝑀

Prove: ∆𝑅𝐻𝑀 ~ ∆𝐺𝐻𝑆

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Reason

Statement

358 | Unit 4

Proof B

Given: 𝑊𝑇 ⊥ 𝑉𝑇 and 𝑊𝑉 ⊥ 𝑅𝑇

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Prove: ∆𝑅𝑇𝑊 ~ ∆𝑇𝑉𝑊

Statement

Reason

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10.

Unit 4 | 359

Proof C

Given: ∠𝑅𝑃𝑇 and ∠𝐶𝑇𝑃 are supplementary.

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Prove: ∆𝑀𝐶𝑇 ~ ∆𝑀𝑅𝑃

Statement

Reason

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360 | Unit 4

Proof D

Given: 𝑊𝑇 ∥ 𝑌𝑆, 𝑊𝑍 ∥ 𝑌𝑅

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Prove: ∆𝑊𝑍𝑇 ~ ∆𝑌𝑅𝑆

Statement

Reason

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Unit 4 | 361

Proof E

Given: : 𝐺𝑀 = 3 𝑅𝑀, 5

𝑅𝐺 = 3 𝐵𝑅, 𝑅𝑀 = 3 𝐵𝑀

Prove: ∆𝑀𝐺𝑅 ~ ∆𝑀𝑅𝐵

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Statement

Reason

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Lesson Summary

When determining if triangles are similar, use definitions, properties, and theorems you have learned throughout this course to gather additional information about the triangles that can be used in a logical sequence to prove triangle similarity. A list of example relationships that may be beneficial is shown. • If the only information given is about lines that are parallel, angle relationships can be derived. • If 2 triangles share an angle, then the angle is congruent to itself by the reflexive property of congruence. • If the lengths of the sides are given, determine if the corresponding sides are proportional. 362 | Unit 4

Practice Problems 1. Triangles 𝐺𝑇𝐴 and 𝑅𝐸𝑇 are shown. Complete the proof. Given: 𝐴𝐺 ∥ 𝑅𝐸

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Prove: ∆𝑇𝐺𝐴 ~ ∆𝑇𝑅𝐸

Since it is given that 𝐴𝐺 ∥ 𝑅𝐸, ∠𝐺𝐴𝑇 ≅ ∠𝑅𝐸𝑇 because if 2 parallel lines are intersected by a transversal, the

are congruent. Also,

∠𝐴𝑇𝐺 ≅ ∠𝐸𝑇𝑅 because

are congruent. .

Therefore, ∆𝑇𝐺𝐴 ~ ∆𝑇𝑅𝐸 by

Complete the proof.

2. Triangle 𝑀𝑅𝑃 is shown.

Given: 𝑀𝐶 = 3 𝑀𝑅 and 𝑀𝐴 = 3 𝑀𝑃 7

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Prove: ∆𝑀𝐶𝐴 ~ ∆𝑀𝑅𝑃

Statement

𝑀𝐶 = 3 𝑀𝑅 and 𝑀𝐴 = 3 𝑀𝑃 7

𝑀𝐶 = 3 and 𝑀𝐴 = 3 7 7 𝑀𝑅 𝑀𝑃

Reason Given Division property of equality Substitution property

∠𝑀 ≅ ∠𝑀

∆𝑀𝐶𝐴 ~ ∆𝑀𝑅𝑃

Unit 4 | 363

3. Triangle 𝐴𝐶𝑊 is shown. Complete the proof. Given: 𝑊𝐴 ≅ 𝐶𝐴 , 𝑅𝑌 ⊥ 𝑊𝐶, and 𝑋𝐺 ⊥ 𝑊𝐶 Prove: ∆𝑅𝑌𝑊 ~ ∆𝑋𝐺𝐶

Reason

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Statement

Given

𝑊𝐴 ≅ 𝐶𝐴 𝑊𝐴 = 𝐶𝐴

Definition of isosceles triangles

∆𝑊𝐴𝐶 is isosceles.

Definition of isosceles triangles

𝑅𝑌 ⊥ 𝑊𝐶

Definition of congruence Given

Definition of perpendicular lines

𝑋𝐺 ⊥ 𝑊𝐶

Given

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∠𝑅𝑌𝑊 is a right angle. 𝑚∠𝑅𝑌𝑊 = 90°

∠𝑋𝐺𝐶 is a right angle.

Definition of perpendicular lines

𝑚∠𝑋𝐺𝑉 = 90°

Definition of congruence ∆𝑅𝑌𝑊 ~ ∆𝑋𝐺𝐶 364 | Unit 4

Review Problems

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4. Triangle 𝐴𝐵𝐶 is dilated. The image is 𝐴′𝐵′𝐶′, find the value of 𝑥.

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5. 𝑊𝑋𝑌𝑍 is a kite, where 𝑚∠𝑊𝑋𝑌 = (11𝑥 + 17)°, 𝑚∠𝑍𝑌𝑋 = 64°, and 𝑚∠𝑍𝑊𝑌 = (7𝑥 + 5)°. Find 𝑚∠𝑍𝑊𝑌.

Unit 4 | 365

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Unit 4, Lesson 14: Solving Problems Using Similarity Warm-Up: Vegetable Garden

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These are the plans for a vegetable garden that a school is designing. Scale: 1 unit = 2.8 feet (ft.)

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Write at least 3 equivalent ratios or equations using lengths from both the diagram and the full-size garden.

Unit 4 | 367

Guided Activity: Similar Figures

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1. Triangles 𝐶𝐾𝑉 and 𝑇𝑁𝑅 are shown, where ∆𝐶𝐾𝑉 ~ ∆𝑇𝑁𝑅.

a. Some measurements of the angles of ∆𝐶𝐾𝑉 and ∆𝑇𝑁𝑅 are given.

𝑚∠𝑇𝑅𝑁 = 28°, 𝑚∠𝑉𝐶𝐾 = (8𝑥 + 26)°, and 𝑚∠𝑅𝑇𝑁 = (12𝑥 − 8)°.

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Determine each value.

𝑥=

𝑚∠𝐶𝐾𝑉 =

Some measurements of the sides of ∆𝐶𝐾𝑉 and ∆𝑇𝑁𝑅 are given.

𝑉𝐾 = 8, 𝐶𝐾 = 5𝑧 + 1.7, 𝐶𝑉 = 5.4, 𝑇𝑅 = 2𝑦 − 1.25, 𝑅𝑁 = 10, and 𝑇𝑁 = 5.25. 368 | Unit 4

b. What is the scale factor from ∆𝐶𝐾𝑉 to ∆𝑇𝑁𝑅?

𝑦=

𝑧=

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c. Determine the value of each variable.

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d. Determine the measure of each side.

𝐶𝐾 =

𝑇𝑅 =

Unit 4 | 369

2. Triangles 𝐾𝑉𝑅 and 𝐾𝐶𝑁 are shown, where ∆𝐾𝑉𝑅 ~ ∆𝐾𝐶𝑁. Some measurements of the sides are given.

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𝑅𝑉 = 4.6, 𝐾𝑅 = 3, 𝐾𝑉 = 5𝑥 − 11, 𝑁𝐶 = 11.5, and 𝐾𝐶 = 3𝑥 + 1. Maria used the information to determine the length of 𝐾𝐶. Her work is shown.

Maria’s Work

𝑁𝐶 = 11.5 = 2.5 𝑅𝑉 4.6

First, Maria found the scale factor.

Then, she used the scale factor to write an equation and solved for 𝑥.

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Finally, she substituted 𝑥 into an equation.

5𝑥 − 11 = 2.5(3𝑥 + 1) 5𝑥 − 11 = 7.5𝑥 + 2.5 −2.5𝑥 = 13.5 𝑥 = −5.4 𝐾𝐶 = 3(−5.4) + 1 𝐾𝐶 = −15.2

Maria knows side lengths cannot have negative values, but she does not know what she did wrong. a. Find and circle Maria’s error.

b. Write a note to Maria explaining what her mistake was and how she can learn from the mistake.

370 | Unit 4

Collaborative Activity: Similar Figures in the Real World

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1. A moving company creates boxes in the shape of rectangular prisms. The company uses a computer program to calculate similar-size box dimensions by using equations. The diagram shows the rectangular bases, 𝐴𝑅𝑉𝐺 and 𝑊𝐾𝐸𝐷, of two different-size boxes. 𝑊𝐷 = 2 in., 𝐴𝑅 = (7𝑤 − 30) in., 𝑅𝑉 = 2.5 in., and 𝑊𝐾 = (6𝑤 − 26) in. a. What is the value of 𝑤?

b. What is the area of 𝐴𝑅𝑉𝐺? Include units.

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c. What is the perimeter of 𝑊𝐾𝐸𝐷? Include units.

The moving company created a third similar rectangular box. The base, 𝑀𝐻𝑆𝑇, is shown, where 𝑀𝐻 = 6.25 in. d. What is the length of 𝑀𝑇? Include units.

Unit 4 | 371

Lesson Summary When 2 figures are similar, there are many equivalent ratios between and within the triangles. Those relationships can be used to find missing lengths. For example, if ∆𝐴𝐵𝐶 is similar to ∆𝐷𝐸𝐹, then pairs of corresponding side lengths are in the same proportion. 𝐴𝐵 = 𝐵𝐶 = 𝐴𝐶 𝐷𝐸 𝐸𝐹 𝐷𝐹

• • •

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It is also known that pairs of side lengths in 1 triangle are in the same proportion as pairs of side lengths in the other triangle. 𝐴𝐵 = 𝐷𝐸 𝐵𝐶 𝐸𝐹 𝐴𝐵 = 𝐷𝐸 𝐴𝐶 𝐷𝐹 𝐴𝐶 = 𝐷𝐹 𝐵𝐶 𝐸𝐹

These equivalent ratios can be used to find unknown side lengths, which is useful in creating and solving equations based on given information. Which equivalent ratios would work to find 𝐷𝐸 and 𝐷𝐹?

• The equivalent ratio 𝐴𝐵 = 𝐷𝐸 can be used to find 𝐷𝐸 because 3 = 𝐷𝐸 , which gives 4 𝐵𝐶 𝐸𝐹 6 𝐷𝐸 = 4.5.

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• The ratio 𝐴𝐵 = 𝐵𝐶 can also be used to find 𝐷𝐸. In this case, 3 = 4 , which also gives 6 𝐷𝐸 𝐸𝐹 𝐷𝐸 𝐷𝐸 = 4.5.

Practice Problems

1. Quadrilateral 𝐴𝐵𝐶𝐷 is similar to quadrilateral 𝐴′𝐵′𝐶′𝐷′. Select all statements that must be true.

□ 𝐴′𝐵′ = 𝐴′𝐶′ 𝐴𝐵 𝐴𝐶

𝐴𝐷 = 𝐵𝐶 □ 𝐴′𝐷′ 𝐵′𝐶′

𝐵𝐷 = 𝐶′𝐷′ □ 𝐵′𝐷′ 𝐶𝐷

□ □

𝐴𝐵 = 𝐴′𝐵′ 𝐶𝐷 𝐶′𝐷′ 𝐵𝐶 = 𝐵′𝐶′ 𝐴′𝐷′ 𝐴𝐷

372 | Unit 4

2. 𝐺𝑀𝑅𝐾 and 𝑃𝑆𝐻𝐷 are shown, where 𝐺𝑀𝑅𝐾 ~ 𝑃𝑆𝐻𝐷. Some measurements of the sides of 𝐺𝑀𝑅𝐾 and 𝑃𝑆𝐻𝐷 are given. 𝐾𝐺 = 12, 𝐺𝑀 = 4𝑥 − 12, 𝑀𝑅 = 7𝑘 − 15, 𝐷𝑃 = 9.6, 𝑆𝑃 = 12.8, 𝐾𝑅 = 30, and 𝑆𝐻 = 4𝑘 − 4.

𝑥=

𝑘=

𝐺𝑀 =

b. Determine the value of each side.

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a. Determine the value of each variable.

𝐷𝐻 =

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c. What is the perimeter of 𝐺𝑀𝑅𝐾?

𝑆𝐻 =

Some measures of the angles of 𝐺𝑀𝑅𝐾 and 𝑃𝑆𝐻𝐷 are given.

𝑚∠𝐾𝐺𝑀 = (11𝑤 − 12)°, 𝑚∠𝐺𝑀𝑅 = 110°, 𝑚∠𝐺𝐾𝑅 = (6𝑧 + 18)°, 𝑚∠𝐷𝑃𝑆 = (8𝑤 + 24)°, and 𝑚∠𝑃𝐷𝐻 = (8𝑧 − 2)°.

d. Determine the value of each variable.

𝑤=

𝑧=

𝑚∠𝐾𝐺𝑀 =

𝑚∠𝐷𝐻𝑆 =

e. Determine the measure of each angle.

𝑚∠𝑃𝐷𝐻 =

Unit 4 | 373

3. Gloria, a landscape engineer, created a trapezoidal garden with a similar trapezoidal garden inside. A diagram of the gardens, 𝑆𝑊𝐶𝑃 and 𝐻𝑇𝐶𝑅, is shown, where 𝑃𝐶 = (3𝑦 + 4) feet (ft.), 𝑅𝐶 = (𝑦 + 3) ft., 𝑆𝑃 = 8.2 ft., 𝑆𝑊 = (7𝑥 − 13) ft., 𝐻𝑇 = (2𝑥 − 2) ft., and 𝐻𝑅 = 4.1 ft.

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a. What is the value of 𝑥?

c. What is the value of 𝑦?

b. What is the length of side 𝑆𝑊? Include units.

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d. What is the length of side 𝑅𝐶? Include units.

e. If 𝑆𝑊 = 𝑊𝐶, what is the perimeter, in feet, of the larger garden?

After Gloria created the gardens, she decided to check the angle measures of the corners. She had the following information written down. 𝑚∠𝑆𝑊𝐶 = (12𝑧 + 6)°, 𝑚∠𝐻𝑇𝐶 = (7𝑧 + 41)°, 𝑚∠𝑇𝐶𝑅 = 90°, 𝑚∠𝑆𝑃𝐶 = (5𝑤 + 11)°, and 𝑚∠𝐻𝑅𝐶 = (4𝑤 + 24)°. f. What is the value of 𝑧?

374 | Unit 4

g. What is the value of 𝑤?

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h. What is 𝑚∠𝑅𝐻𝑇?

Review Problems

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B. 4

A. 3.5

4. Segment 𝐴′𝐵′ is parallel to 𝐴𝐵. What is the length of 𝐵𝐵′?

C. 10

D. 10.5

5. Elena thinks length 𝐵𝐶 is 16.5 units. Lin thinks the length of 𝐵𝐶 is 17.1 units. Do you agree with either of them? Explain or show your reasoning.

Unit 4 | 375

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Unit 5: Right Triangles and Trigonometry

Unit 5 | 377

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Unit 5, Lesson 1: Equivalent Radical Expressions Warm-Up: Error Analysis Leilana and Khamari wrote different numerical expressions and expanded them to show that each has a value of 16. Their work is shown. 24 = 2 ∙ 2 ∙ 2 ∙ 2 = 16

Khamari’s Work

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Leilana’s Work

28 = 2 ∙ 8 = 16

1. Explain whether you agree with Leilana’s work or Khamari’s work.

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2. Write another numerical expression involving exponents with a value of 16.

Unit 5 | 379

Exploration Activity: Equivalent Radical Expressions Ameer was asked to write an equivalent expression for expression are shown.

18 . His work and resulting

Ameer’s Work

Equivalent Expression

18 = 6 ∙ 3 = 6 ∙ 3

6∙ 3

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Given Expression

Sophia wrote a different equivalent expression for expression are shown. Given Expression

18 . Her work and resulting

Sophia’s Work

2∙ 9

18 = 2 ∙ 9

Equivalent Expression 3 2

1. What similarities and differences do you notice between Ameer’s work and Sophia’s work?

Differences

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Similarities

380 | Unit 5

2. With your partner, complete the work to find equivalent expressions for different ways.

Given Expression

175 two

Equivalent Radical Expression

Work

175 = 35 ∙ 5 =

∙

175

175 = 25 ∙ 7 =

∙

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175

3. Which equivalent radical expression above can be rewritten in the form of an integer multiplied by a radical? Use it to complete the statement. because

is a perfect square; therefore,

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∙

Unit 5 | 381

Guided Activity: Equivalent Radical Expressions Recall the power of a product law of exponents. It states that the product of 2 or more numbers raised to a power is equal to the product of each number raised to the same power, as shown algebraically. or 𝑎𝑚 ∙ 𝑏𝑚 = (𝑎𝑏)𝑚

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(𝑎𝑏)𝑚 = 𝑎𝑚 ∙ 𝑏𝑚

The product law of radicals works similarly. Each of the equivalent expressions explored previously demonstrates the product law of radicals. 18 = 6 ∙ 3 18 = 9 ∙ 2

175 = 35 ∙ 5 175 = 25 ∙ 7

325 in 2 different ways using the product law of radicals.

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1. Rewrite

When a radical expression is rewritten as a product of its factors, each of the radicals will have the same index.

When rewriting radical expressions, mathematicians generally try to rewrite the expression in the form of an integer multiplied by a radical when possible. Two examples are shown. 18 = 9 ∙ 2, which can be rewritten as 3 2

175 = 25 ∙ 7, which can be rewritten as 5 7 2. What was done to rewrite the example expressions?

382 | Unit 5

3. The following radical expressions are equivalent to 3

32 ∙ 10

64 ∙ 5

320.

8 ∙ 40

16 ∙ 20

a. Circle the expressions that can be rewritten in the form of an integer multiplied by a radical.

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b. Discuss with your partner why these expressions can be rewritten.

c. Rewrite the expressions in the form of an integer multiplied by a radical.

4. Some radical expressions are given as an integer multiplied by the radical before any rewriting is done. Priya and Raj were asked to rewrite this type of expression. Their work is shown.

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Priya’s Work Raj’s Work

3 20 = 3 ∙ 4 ∙ 5 = 3 ∙ 2 5 = 6 5 3 20 = 3 ∙ 4 ∙ 5 = 3 ∙ 2 5 = 5 5

Explain whether Priya or Raj rewrote the expression correctly.

Unit 5 | 383

Lesson Summary Numerical expressions of irrational and rational numbers involving radicals can be rewritten using the product law of radicals as shown. 𝑚

𝑚

𝑎𝑏 = 𝑎 ∙ 𝑏 or

𝑚

𝑎 ∙ 𝑏 = 𝑎𝑏

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When rewriting radical expressions, mathematicians generally try to rewrite the expression in the form of an integer multiplied by a radical when possible. An example is shown. 3 112 = 3 ∙ 16 ∙ 7 = 3 ∙ 4 ∙ 7 = 12 7

• Notice the radicand, 112, was rewritten as the product of 16 and 7. • Since 16 is a perfect square,

16 resulted in the integer 4.

Practice Problems

• When multiplying all the remaining factors, 3 ∙ 4 ∙ 7 = 12 7.

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1. Write 2 equivalent expressions for each radical expression. Show all your steps. Given Expression

Equivalent Expressions

135 =

𝟏𝟑𝟓

𝟔𝟎𝟎

384 | Unit 5

135 =

□ 8∙5 □ 2 20 □ 4 ∙ 10 □ 4 5 □ 5 8 □ 2 10 □ 2 ∙ 20

40 .

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2. Select all the expressions that are equivalent to

3. Write an equivalent expression for 5 56 .

378 in the form of an integer multiplied by a

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4. Write an equivalent expression for radical.

Review Problems

5. Pentagon 𝐴′𝐵′𝐶′𝐷′𝐸′ is the image of pentagon 𝐴𝐵𝐶𝐷𝐸 after a dilation centered at 𝐹. What is the scale factor of this dilation?

Unit 5 | 385

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6. In isosceles triangle 𝐷𝐴𝐶, 𝐴𝐷 is congruent to 𝐴𝐶, and 𝐴𝐵 is an angle bisector of ∠𝐷𝐴𝐶. How does Kiran know that 𝐴𝐵 is a perpendicular bisector of 𝐶𝐷?

386 | Unit 5

Unit 5, Lesson 2: Adding and Subtracting Radical Expressions Warm-Up: Adding and Subtracting Algebraic Expressions 1. Find the sum or difference of each algebraic expression. −7𝑡 − 40𝑡

−35𝑏 + 10𝑏

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4𝑥 + 23𝑥

2. Complete the statement to explain why it is possible to combine the terms in each of these expressions. terms can be

When adding and subtracting algebraic expressions, combined.

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□ 17 + 32 □ 5 13 − 8 3 □ −7 17 + 3 17 □ 9 5+ 5 □ 8 40 − 4 9 □ 11 48 − 6 48 □ −2 49 − 3 4

3. Which, if any, of the following radical expressions do you think can be combined?

Unit 5 | 387

Exploration Activity: Adding and Subtracting Radical Expressions 1. Determine the value of each expression. 25 + 100

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25 + 100

125.

2. Explain which of the expressions is equivalent to

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3. Paloma says 16 − 9 = 7. Ty’rique says 16 − 9 = 1. Who is correct? Verify by entering 16 − 9 into a calculator. Then, complete the statement.

16 − 9 is equal to

is correct because

4. Xavier says 12 + 2 12 = 3 12. Sam says 12 + 2 12 = 2 24. Who is correct? Verify using a calculator. Then, complete the statement.

is correct because approximately equal to 388 | Unit 5

12 + 2 12 and

are both

Guided Activity: Adding and Subtracting Radical Expressions When adding and subtracting algebraic expressions, only like terms can be combined. Adding and subtracting radical expressions is similar because only like radicals can be combined. 1. Find the sum or difference of each radical expression.

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a. 4 5 + 7 5 = 3

b. 30 2 − 23 2 = 3

c. −10 7 − 23 7 = d. − 17 + 17 =

Three steps can be used when adding and subtracting radical expressions.

Step 1: Find perfect squares for square roots or perfect cubes for cube roots that are factors of the radicand.

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Step 2: Rewrite the radical expressions by applying the product law of radicals and evaluating any perfect squares or cubes. Step 3: Add or subtract like radical terms. 2. Use the steps to add the radical expression shown. 3 6 + 5 24

Step 1: Both terms include square roots. If possible, find perfect squares that are factors of the radicands, 6 and 24. Step 2: Rewrite the radical expressions by applying the product law of radicals and evaluating any perfect squares. 3 6 + 5 24 =

+ Unit 5 | 389

3. Three students added Gerald’s Work

180 and

45 differently. Their work is shown.

180 + 45

Kalo’s Work

Paola’s Work

180 + 45

3 20 + 3 5

2 45 + 45 3 45

6 5+3 5 9 5

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a. Use a calculator to verify if the students’ answers are equivalent.

b. Why did the students end up with different expressions for their sums?

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c. Show how Gerald and Kalo could continue their work to arrive at the same expression as Paola.

Kalo’s Next Steps

3 45

3 20 + 3 5

Gerald’s Next Steps

390 | Unit 5

4. Megh found a sum of cube roots, but she made an error. Her work is shown. 3

32 + 64 = 8 ∙ 4 + 4 = 2 4 + 4 = 6 4

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Explain the error Megh made.

Collaborative Activity: Adding and Subtracting Radical Expressions

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□ 17 + 32 □ 5 13 − 8 3 □ −7 17 + 3 17 □ 9 5+ 5 □ 8 40 − 4 9 □ 11 48 − 6 48 □ −2 49 − 3 4

1. Select all the expressions that can be combined. For the selected expressions, combine the terms to find the sum or difference. Show your work.

2. Work with your partner to find the sum and/or difference of the following problems. If you cannot add or subtract, circle the expression. a.

6+3 6

20 − 3 45

Unit 5 | 391

75 + 2 48

72 + 18 − 15

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Lesson Summary

When adding and subtracting algebraic expressions, only like terms can be combined. Adding and subtracting radical expressions is similar because only like radicals can be combined. Like radicals are radical terms that have the same index and the same radicand.

In the expression, 15 , the index is 3 and the radicand is 15. This expression can be 3 combined with 4 15 because both expressions have an index of 3 and a radicand of 15. 3

15 + 4 15 = 5 15

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15 − 4 15 = −3 15

Practice Problems

1. Find each sum or difference. a. 8 24 + 11 96

392 | Unit 5

56 − 54

2. Find each sum or difference. 3

27,000 − 512

6 − 48

50 + 18 + 10

4 − 5 8 + 98

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3. Michaela was finding the sum of the expression 63 + 6 28 , but she made an error in her work. Michaela’s work is shown.

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Describe and correct the error Michaela made in her work.

Michaela’s Work 63 + 6 28

9∙7+6 7∙4

9 7+6∙4 7 9 7 + 24 7 33 7

Review Problems

4. Segment 𝐴′𝐵′ is parallel to segment 𝐴𝐵. a. What is the length of segment 𝐴′𝐴?

Unit 5 | 393

Lines 𝐵𝐶 and 𝐷𝐸 are both vertical. What is the length of 𝐴𝐷? A. 4.5 B. 5

C. 7.5

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D. 10

5. Match each vocabulary term with its definition.

C. Dilation

2. When 2 angles of one triangle are congruent to 2 angles of a second triangle, the 2 triangles will be similar.

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D. Similar figures

B. Scale factor

1. There is a sequence of rigid motions and dilations that takes the first figure onto the second.

A. Scale drawing

E. Angle-angle similarity theorem

3. A transformation using a center 𝐶 and scale factor 𝑘 that takes a point 𝐴 to another point along the ray 𝐶𝐴 whose distance is 𝑘 times farther from 𝐶 than 𝐴 is.

4. A drawing in which all lengths in the drawing correspond to lengths in the object by the same scale.

394 | Unit 5

5. A constant multiple by which all lengths of the original figure are multiplied.

Unit 5, Lesson 3: Multiplying and Dividing Radical Expressions Warm-Up: True Equations

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1. Using the digits 0 to 9, place a digit in each blue box to make both equations true. Each digit can only be used once.

Unit 5 | 395

Guided Activity: Multiplying and Dividing Radical Expressions When dividing radicals with the same index, the quotient law of radicals is used to rewrite the expression. Each of the following equivalent expressions demonstrates the quotient law of radicals.

2 2

= 21 = 1 = 1 21

using the quotient law of radicals.

1. Rewrite 3

= 13 = 1 = 1

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When dividing radical expressions, rewrite radical expressions in equivalent forms, and use the quotient law to eliminate radicals where possible.

125

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2. Write an equivalent expression for

Equivalent Expression

245

by completing the steps in the table.

125

245

∙5 ∙5

∙ 5 ∙ 5

396 | Unit 5

Step Original expression

Rewrite the radicands, and products of a factor and 5.

, as the

Rewrite the radicals using the law of radicals.

Evaluate each perfect square root. Because

5 = 1, 5

∙ 5 ∙ 5 =

3. With your partner, rewrite the expression

275 44

using the quotient laws of radicals.

8 ∙ 8. His explanation is

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4. Marcus was asked to explain how he would multiply shown.

I would use the product law of radicals to multiply the radical expressions and then use the law again to rewrite the square root. 8 ∙ 8 = 8 ∙ 8 = 64 = 8

12 ∙ 12 using the product law of radicals.

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5. Multiply

Explain whether you agree that his expression has a value of 8.

6. Multiply

6 ∙ 18 .

Unit 5 | 397

Collaborative Activity: Pass Around In groups of 4, identify who students A, B, C, and D are. Complete the first step of your problem. Then, pass your problem to the student on your right to complete the next step. For each new step, continue passing the problems to the student on your right. Student A

Student B 3

14 ∙ 20

625

200

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Student C

256

108

24 ∙ 30

Student D

398 | Unit 5

Lesson Summary When multiplying radicals, the coefficients of the radicals are multiplied together, and the radicands are multiplied together. Then, the expression is simplified if possible. An example of multiplying

15 ∙ 7 6 is shown.

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15 ∙ 7 6 = 7 ∙ 15 ∙ 6 = 7 90 = 7 9 ∙ 10 = 7 ∙ 3 10 = 21 10

Similar strategies can be applied when dividing radical expressions.

In cases in which the resulting expression has a fraction with a radical in the denominator, mathematicians use a technique called rationalizing the denominator to rewrite an expression. An example of how to rationalize the denominator for the expression 2

∙

30 2

30 2 2 = 15 2

Because of the identity property of multiplication, the expression 2

because

2 2

is shown.

can be multiplied

= 1. This allows for the denominator to be rewritten as a rational

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number while maintaining equivalence with the original expression because 2 ∙ 2 = 4 = 2.

Practice Problems

1. Anjelica rewrote the expression 3 264, but she made an error. Her work is shown. 3

8 ∙ 33

2 33 = 3 = 3 9 81 9∙9 264

a. Explain the error Anjelica made.

Unit 5 | 399

2. Determine each product or quotient.

9∙ 9

56 77

3. Determine each product or quotient. 3

8 42 2 7

a. 5 6 ∙ 4 4

c. 2 22 8

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Review Problems

192

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4. Write an equivalent expression for each radical expression. c.

108

5. Right triangle 𝑀𝐶𝑊 is shown, where 𝐶𝑊 = 6 units and 𝑀𝑊 = 7.8 units. What is the length of 𝐶𝑀? Round to the nearest hundredth.

400 | Unit 5

Unit 5, Lesson 4: Proving the Pythagorean Theorem

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Warm-Up: Notice and Wonder: Variable Version

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What do you notice? What do you wonder?

Unit 5 | 401

Exploration Activity: Prove Pythagoras Right A right triangle is shown with its sides labeled. Elena is working with the equivalent ratios she wrote in the Warm-Up. She rewrites 𝑎 = 𝑐 as 𝑥

𝑎

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𝑎2 = 𝑥𝑐. Diego notices and comments, “I got 𝑏2 = 𝑦𝑐. The 𝑎2 and 𝑏2 remind me of the Pythagorean theorem.” Elena says, “The Pythagorean theorem says that 𝑎2 + 𝑏2 = 𝑐2. I bet we could figure out how to show that.”

1. Explain how Elena got from 𝑎 = 𝑐 to 𝑎2 = 𝑥𝑐. 𝑥

𝑎

2. Which equivalent ratios of side lengths did Diego use to get 𝑏2 = 𝑦𝑐? 3. Use the segment addition postulate to complete the statement. +

𝑐=

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4. Complete the table to prove 𝑎2 + 𝑏2 = 𝑐2 in a right triangle with legs with lengths 𝑎 and 𝑏 and hypotenuse with length 𝑐. Description

Equation

Elena’s equation

𝑎2 = 𝑥𝑐

Diego’s equation

Add the 2 equations together. +

402 | Unit 5

𝑏2 = 𝑦𝑐

= 𝑐( =

= 𝑥𝑐 + 𝑦𝑐 +

(

)

Guided Activity: The Pythagorean Theorem and Triangle Similarity

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1. Triangle 𝑅𝑊𝐻 is shown, where 𝑊𝐵 is perpendicular to 𝐵𝐻, and 𝑊𝑅 is perpendicular to 𝑊𝐻.

a. Complete the statements.

congruent similar

∆𝑅𝐵𝑊 ∆𝐻𝑊𝑅

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∆𝐻𝐵𝑊 is

∆𝑊𝐵𝐻 ∆𝐻𝑊𝑅

congruent similar

𝑊𝐵 divides right triangle 𝑅𝑊𝐻 into 2

AA ASA SSS

triangles such that ∆𝑊𝐵𝑅 is

congruence. similarity. AA ASA SSS

congruence. similarity.

b. Use the labels to show that ℎ2 + 𝑚2 = 𝑣2.

Unit 5 | 403

Collaborative Activity: An Alternate Approach

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When Pythagoras proved his theorem, he used the 2 images shown here. Can you figure out how he used these diagrams to prove 𝑎2 + 𝑏2 = 𝑐2 in a right triangle with hypotenuse length 𝑐?

404 | Unit 5

Lesson Summary In any right triangle with legs 𝑎 and 𝑏 and hypotenuse 𝑐, it is known that 𝑎2 + 𝑏2 = 𝑐2. This is called the Pythagorean theorem. But why does it work?

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An altitude can be drawn from the hypotenuse of a right triangle to prove the Pythagorean theorem.

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The angle-angle (AA) triangle similarity conditions can be used to show that all 3 triangles created are similar. Because the triangles are similar, corresponding side lengths are proportional.

• Because the largest triangle is similar to the smallest triangle, 𝑐 = 𝑎 . •

𝑎 𝑑 𝑐 𝑏 Because the largest triangle is similar to the middle triangle, = . 𝑏 𝑒

• These equations can be rewritten as 𝑎2 = 𝑐𝑑 and 𝑏2 = 𝑐𝑒.

• When the 2 equations are added, the result is 𝑎2 + 𝑏2 = 𝑐𝑑 + 𝑐𝑒 or 𝑎2 + 𝑏2 = 𝑐(𝑑 + 𝑒).

• From the original diagram, it can be determined that 𝑑 + 𝑒 = 𝑐, so 𝑎2 + 𝑏2 = 𝑐(𝑐) or 𝑎2 + 𝑏2 = 𝑐2. Using the Pythagorean theorem, a triangle’s angles can be described without ever drawing it. • A triangle with side lengths 8, 15, and 17 is right because 172 = 82 + 152.

• A triangle with side lengths 8, 15, and 18 is obtuse because 182 > 82 + 152. • A triangle with side lengths 8, 15, and 16 is acute because 162 < 82 + 152. © Accelerate Learning Inc. - All Rights Reserved

Unit 5 | 405

Practice Problems 1. Which of the following are right triangles? A. Triangle 𝐴𝐵𝐶 with 𝐴𝐶 = 6, 𝐵𝐶 = 9, and 𝐴𝐵 = 12

B. Triangle 𝐷𝐸𝐹 with 𝐷𝐸 = 8, 𝐸𝐹 = 10, and 𝐹𝐷 = 13

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C. Triangle 𝐺𝐻𝐼 with 𝐺𝐼 = 9, 𝐻𝐼 = 12, and 𝐺𝐻 = 15

D. Triangle 𝐽𝐾𝐿 with 𝐽𝐿 = 10, 𝐾𝐿 = 13, and 𝐽𝐿 = 17

2. In right triangle 𝐴𝐵𝐶, a square is drawn on each of its sides. An altitude, 𝐶𝐷, is drawn to the hypotenuse 𝐴𝐵 and extended to the opposite side of the square on 𝐹𝐸. In class, we discussed Elena’s observation that 𝑎2 = 𝑥𝑐 and Diego’s observation that 𝑏2 = 𝑦𝑐. Mai observes that these statements can be thought of as claims about the areas of rectangles.

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a. Which rectangle has the same area as 𝐵𝐺𝐻𝐶? b. Which rectangle has the same area as 𝐴𝐶𝐼𝐽?

3. Andre says he can find the length of the third side of triangle 𝐴𝐵𝐶 and it is 5 units. Mai disagrees and thinks that the side length is unknown. Do you agree with either of them? Show or explain your reasoning.

406 | Unit 5

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4. Similar triangles 𝑅𝐿𝑁, 𝐷𝑅𝑁, and 𝐷𝐿𝑅 are shown.

Complete the steps to show that (ℎ + 𝑧)2 = 𝑝2 + 𝑤2.

2 2

) 2

(

= ℎ (

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Review Problems

(

𝑧+ℎ

= 𝑧

𝑧+ℎ

5. In right triangle 𝐴𝐵𝐶, altitude 𝐶𝐷 is drawn to its hypotenuse. Find 2 triangles which must be similar to ∆𝐴𝐵𝐶.

Unit 5 | 407

6. In right triangle 𝐴𝐵𝐶, altitude 𝐶𝐷 with length 6 is drawn to its hypotenuse. We also know 𝐴𝐷 = 12. What is the length of 𝐷𝐵? A. 1 2

B. 3 C. 4

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D. 6

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7. Priya is trying to determine if ∆𝐴𝐷𝐶 is congruent to ∆𝐶𝐵𝐴. She knows that segments 𝐴𝐵 and 𝐷𝐶 are congruent. She also knows that ∠𝐷𝐶𝐴 and ∠𝐵𝐴𝐶 are congruent. Does she have enough information to determine that the triangles are congruent? Explain your reasoning.

408 | Unit 5

Unit 5, Lesson 5: The Converse of the Pythagorean Theorem Warm-Up: Notice and Wonder: Right Triangles

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What do you notice? What do you wonder?

Unit 5 | 409

Exploration Activity: The Converse of the Pythagorean Theorem

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1. Line segments 𝑇𝑆, 𝑌𝐻, 𝐵𝑊, 𝑀𝑃, 𝑋𝐹, and 𝐾𝑅 are shown.

a. Trace each line segment onto a piece of tracing paper. Cut around each line segment so that you have 6 separate pieces of tracing paper.

b. Complete the table by manipulating the given set of line segments to determine if they can be used to create a right triangle. Can the set of line segments create a right triangle?

Set of Line Segments

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𝑀𝑃, 𝑌𝐻, and 𝑇𝑆

𝑌𝐻, 𝑋𝐹, and 𝑀𝑃

No Yes No Yes No

𝐵𝑊, 𝐾𝑅, and 𝑀𝑃

Yes

c. Discuss with your partner other strategies that could be used to determine if each set of line segments creates a right triangle. Summarize your discussion.

410 | Unit 5

2. A triangle has side lengths of 17 units, 15 units, and 8 units.

a. If the Pythagorean theorem, 𝑎2 + 𝑏2 = 𝑐2, is true, what can be concluded about the triangle?

b. Complete the statement. is is not

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A triangle with side lengths of 17 units, 15 units, and 8 units = ≠

right triangle because 152 + 82

172.

3. A triangle has side lengths of 14 units, 9 units, and 10 units. Complete the statement.

= ≠

is is not

a right

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triangle because 92 + 102

A triangle with side lengths of 14 units, 9 units, and 10 units

Unit 5 | 411

Collaborative Activity: Is it a Right Triangle? 1. James and Brianna are determining if a triangle with side lengths of 12 units, 13 units, and 5 units is a right triangle. Their work is shown. James’s Work

Brianna’s Work

52 + 132 = 122

122 + 52 = 132

144 + 25 = 169

194 ≠ 144

The triangle is not a right triangle.

169 = 169

The triangle is a right triangle.

Explain whose work is incorrect.

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25 + 169 = 144

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Side Lengths

2. Determine if each set of side lengths forms a right triangle by using the converse of the Pythagorean theorem.

15 inches (in.), 9 in., and 12 in.

8 feet (ft.), 7 ft., and 11 ft. 25 ft., 7 ft., and 24 ft.

24 in., 10 in., and 26 in. 412 | Unit 5

Work

Right Triangle Yes No Yes No Yes No Yes No

Lesson Summary The Pythagorean theorem can be used to determine a missing side length in a right triangle. The converse of the Pythagorean theorem can be used to determine whether a triangle is a right triangle.

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The converse of the Pythagorean theorem states that if the lengths of 𝑎, 𝑏, and 𝑐 of the three sides of a triangle satisfy the relationship 𝑎2 + 𝑏2 = 𝑐2, then the triangle is a right triangle.

Practice Problems

For example, since 82 + 62 = 64 + 36 = 102, any triangle with side lengths of 6, 8, and 10 must be a right triangle. The Pythagorean theorem and its converse can be used to determine if a triangle is a right triangle just by using the side lengths. If 𝑎2 + 𝑏2 = 𝑐2, it is a right triangle, where 𝑐 is the hypotenuse of the triangle. If 𝑎2 + 𝑏2 ≠ 𝑐2, it is not a right triangle, where 𝑐 is the longest side of the triangle.

1. Select all the sets of side lengths that can be used to create a right triangle.

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□ 113 centimeters (cm), 15 cm, and 112 cm □ 101 cm, 120 cm, and 99 cm □ 41 ft., 9 ft., and 40 ft. □ 13 in., 9 in., and 15 in. □ 3 ft., 5 ft., and 4 ft. □ 5 ft., 7 ft., and 6 ft.

2. A triangle has side lengths of 15 units, 8 units, and 17 units. Is the triangle a right triangle? Explain your reasoning.

Unit 5 | 413

3. Kyra incorrectly determined that a triangle with the side lengths of 5 units, 7 units, and 12 units is a right triangle. Her work is shown.

Kyra’s Work 52 + 72 = 122

Describe Kyra’s error, and show the corrected work.

10 + 14 = 24 24 = 24

Review Problems 4. Quadrilaterals 𝑄 and 𝑃 are similar.

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Yes, it forms a right triangle.

B. 3 5

C. 4 5

D. 5

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A. 2

What is the scale factor of the dilation that takes 𝑄 to 𝑃?

5. Select the sequence of transformations of ∆𝐴𝐷𝐸 that would show that ∆𝐴𝐵𝐶 and ∆𝐴𝐸𝐷 are similar. The length of 𝐴𝐶 is 6, or 𝐴𝐶 = 6.

A. Dilate from center 𝐴 by a scale factor of 2, and then reflect over line 𝐴𝐶. B. Dilate from center 𝐴 by a scale factor of 2, and then rotate 60° around angle 𝐴.

C. Translate by directed line segment 𝐷𝐶, and then reflect over line 𝐴𝐶.

D. Dilate from center 𝐴 by a scale factor of 4, and then reflect over line 𝐴𝐶. 414 | Unit 5

Unit 5, Lesson 6: Finding Unknown Values in Right Triangles Warm-Up: Which One Doesn’t Belong: Triangles Which one doesn’t belong? B.

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Unit 5 | 415

Collaborative Activity: Info Gap: Similar Sequence Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the data card:

If your teacher gives you the problem card:

2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)

1. Silently read your card and think about what information you need to answer the question.

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1. Silently read the information on your card.

3. Explain to your partner how you are using the information to solve the problem. 4. When you have enough information, share the problem card with your partner, and solve the problem independently.

3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”

2. Ask your partner for the specific information that you need.

4. Read the problem card, and solve the problem independently.

5. Read the data card, and discuss your reasoning.

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5. Share the data card, and discuss your reasoning.

416 | Unit 5

Guided Activity: Relatively Reasonable

Lesson Summary

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Triangle 𝐴𝐵𝐶 is similar to triangle 𝐴′𝐵′𝐶′. Give reasonable measurements for all 3 sides of triangle 𝐴′𝐵′𝐶′. Explain your reasoning.

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There are multiple strategies for finding unknown side lengths in similar right triangles. Similar right triangles 𝐴𝐵𝐶 and 𝐹𝐺𝐻 are shown. Since ∆𝐴𝐵𝐶 is a right triangle,42 + 𝑥2 = 52.

Since ∆𝐹𝐺𝐻 is a right 2

triangle, 𝑦2 + � 12 � = 42. 5

Since ∆𝐴𝐵𝐶 ~ ∆𝐹𝐺𝐻, there are many equations that can be written using proportional relationships. Any combination of these equations can be used to find the values of 𝑥 and 𝑦. Examples are shown.

• By similarity, 5 (𝑦) = 4, so 𝑦 = 16 . Substituting 𝑦 = 16 into the Pythagorean theorem 5 4 2 2 16 12 2 gives � � + � � = 4 , which is true. 5 5

Unit 5 | 417

Practice Problems

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1. In the right triangles shown, the measure of ∠𝐴𝐵𝐶 is the same as the measure of ∠𝐸𝐵𝐷. What is the length of side 𝐵𝐸?

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2. In right triangle 𝐴𝐵𝐶, ∠𝐶 is a right angle, 𝐴𝐵 = 13, and 𝐵𝐶 = 5. What is the length of 𝐴𝐶?

3. In this diagram, 𝐴𝐶 and 𝐷𝐸 are parallel, and 𝐷𝐶 is perpendicular to each of them. 𝐴𝐶 ∥ 𝐷𝐸, 𝐷𝐶 ⊥ 𝐷𝐸, 𝐷𝐶 ⊥ 𝐴𝐶

What is a reasonable estimate for the length of side 𝐵𝐸? A. 1 3

B. 1 C. 5 D. 5

418 | Unit 5

Review Problems 4. Select all of the right triangles.

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□ Triangle 𝐴𝐵𝐶 with 𝐴𝐵 = 30, 𝐵𝐶 = 40, and 𝐴𝐶 = 50 □ Triangle 𝑋𝑌𝑍 with 𝑋𝑌 = 1, 𝑌𝑍 = 1, and 𝑋𝑍 = 2 □ Triangle 𝐸𝐹𝐺 with 𝐸𝐹 = 8, 𝐹𝐺 = 15, and 𝐸𝐺 = 17 □ Triangle 𝐿𝑀𝑁 with 𝐿𝑀 = 7, 𝑀𝑁 = 24, and 𝐿𝑁 = 25 □ Triangle 𝑄𝑅𝑆 with 𝑄𝑅 = 4, 𝑅𝑆 = 5, and 𝑄𝑆 = 6

5. Andre says he can find the length of the third side of triangle ∆𝐴𝐵𝐶 and it is 13 units. Mai disagrees and thinks that the side length is unknown. Who do you agree with? Show or explain your reasoning.

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6. In right triangle 𝐴𝐵𝐶, altitude 𝐶𝐷 with length ℎ is drawn to its hypotenuse. We also know 𝐴𝐷 = 8 and 𝐷𝐵 = 2. What is the value of ℎ?

Unit 5 | 419

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Unit 5, Lesson 7: Angles and Steepness Warm-Up: Ratios Galore

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Triangle 𝐴𝐵𝐶 is similar to triangle 𝐷𝐸𝐹. Write as many equations as you can to describe the relationships between the sides and angles of the 2 triangles.

Exploration Activity: Can You Calculate?

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Find the values of 𝑥, 𝑦, and 𝑧. If there is not enough information, what else do you need to know?

𝑥=

𝑦=

𝑧=

Unit 5 | 421

Collaborative Activity: Is It Accessible?

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1. Some buildings offer ramps in addition to stairs so people in wheelchairs have access to the building. What characteristics make a ramp safe?

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2. A school has 4 steps to the front door. Each step is 7 inches (in.) tall. Design a ramp for the school.

3. Your teacher will give you the Americans with Disabilities Act (ADA) guidelines. Does your design follow the rules of this law? If not, draw a new design that does.

422 | Unit 5

Lesson Summary Because of the Pythagorean theorem, if any 2 sides of a right triangle are known, then the length of the third side can be calculated. But what if a side and an angle are known rather than 2 sides?

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All right triangles with 1 pair of congruent acute angles are similar by angle-angle (AA) triangle similarity conditions. Knowing just 1 side length in addition to those corresponding angle measures is enough to uniquely define the triangle. The American with Disabilities Act includes guidelines for safe and accessible wheelchair ramps. Ramps must form a maximum 4.8° angle with the ground, which creates a maximum 1 ∶ 12 ratio for the legs of the right triangle.

Assume a ramp for a 3 in. threshold is being built, as shown.

• Similarity can be used to find length 𝑥. By corresponding sides, 1 = 3 , so 𝑥 is 36 12 𝑥 units.

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• To find length 𝑦, the Pythagorean theorem can be used. 32 + 362 = 𝑦2, so 𝑦 = 1,305 or about 36.1 units.

To build a ramp that goes up 3 in., it needs to start 36 in., or 3 feet (ft.), out from the edge of the threshold and use a board that’s about 36.1 in. long.

Unit 5 | 423

Practice Problems 1. The Americans with Disabilities Act states that ramps must have an angle less than or equal to 4.8°. Remember, a 4.8° angle in a right triangle has a 1 ∶ 12 ratio for the legs. Select all ramps that meet the Americans with Disabilities Act requirements.

Triangle D

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Triangle C

Triangle B

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Triangle A

Triangle E

□ Triangle A □ Triangle B □ Triangle C □ Triangle D □ Triangle E

424 | Unit 5

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2. Find the missing side in each triangle using any method. Check your answers using a different method.

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3. The Americans with Disabilities Act states that ramps must have an angle less than or equal to 4.8°. Remember, a 4.8° angle in a right triangle has a 1 ∶ 12 ratio for the legs. Design 2 ramps that meet the Americans with Disabilities Act requirements.

Review Problems

4. In this diagram, 𝐴𝐶 and 𝐷𝐸 are parallel, and 𝐷𝐶 is perpendicular to each of them. If 𝐵𝐷 has length 4 , calculate the length of side 𝐷𝐸. A. 1 3

B. 1

𝐴𝐶 ∥ 𝐷𝐸, 𝐷𝐶 ⊥ 𝐷𝐸, 𝐷𝐶 ⊥ 𝐴𝐶

C. 3 D. 6

Unit 5 | 425

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5. Lin says she has memorized the lengths of a few right triangles, for example, 3, 4, and 5. She is trying to compile a list of several right triangles but needs your help. Find the lengths of at least 2 triangles that are right.

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6. In triangle 𝐴𝐵𝐶, the measure of angle 𝐴 is 35° and the measure of angle 𝐵 is 20°. In triangle 𝐷𝐸𝐹, the measure of angle 𝐷 is 35° and the measure of angle 𝐹 is 125°. Are triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 similar? Explain or show your reasoning.

426 | Unit 5

Unit 5, Lesson 8: Half a Square Warm-Up: Diagonals of Rectangles

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Calculate the values of 𝑥 and 𝑦.

Unit 5 | 427

Exploration Activity: Decomposing Squares 1. Draw a square with side lengths of 1 centimeter (cm).

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a. Estimate the length of the diagonal.

b. Calculate the length of the diagonal.

2. Measure the side length and diagonal length of several squares, in cm. Compute the ratio of side to diagonal length for each.

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The triangles created in the squares with the diagonal are 45 − 45 − 90 triangles. 3. The table shows 3 different 45 − 45 − 90 triangles. ∆𝑾𝑩𝑵

∆𝑨𝒁𝑷

∆𝑲𝑩𝑻

428 | Unit 5

a. Discuss with your partner the relationship between the legs of a 45 − 45 − 90 triangle and its hypotenuse. Write a conjecture.

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b. Ask 2 other groups for the conjectures they wrote. Listen to each group’s conjecture and write a summary. You are the only person who should write in the first column of the table. Have a person from each of the other groups initial next to your summary, stating that your summary is correct. Initials

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Conjecture

4. An isosceles right triangle is shown.

Describe how the triangle shows the relationships in a 45 − 45 − 90 triangle.

Unit 5 | 429

Guided Activity: Generalize Half Squares

𝑅𝑄 =

𝑃𝑅 =

𝑌𝑍 =

𝑋𝑌 =

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𝐴𝐶 =

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Calculate the lengths of the 5 unlabeled sides.

430 | Unit 5

Lesson Summary Drawing the diagonal of a square decomposes the square into 2 congruent triangles. They are right isosceles triangles with acute angles of 45°. These congruent angles make all right isosceles triangles similar by the angle-angle (AA) triangle similarity conditions.

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Consider an isosceles right triangle with legs 1 unit long, where 𝑐 is the length of the hypotenuse. By the Pythagorean theorem, 12 + 12 = 𝑐2, so 𝑐 = 2. The hypotenuse of an isosceles right triangle with legs 1 unit long is 2 units long. Now, consider an isosceles right triangle with legs 𝑥 units long. By the AA triangle similarity, the triangle is similar to the isosceles right triangle with side lengths of 1, 1, and 2 units. A scale factor of 𝑥 takes the triangle with a leg length of 1 to the triangle with a leg length of 𝑥. Therefore, the hypotenuse of the isosceles right triangle with legs 𝑥 units long is 𝑥 2 units long.

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Two examples of 45 − 45 − 90 triangles are shown.

In ∆𝐴𝐵𝐶, 𝑥 = 6, so 𝐴𝐶 is 6 units long and 𝐵𝐶 is 6 2 units long.

In ∆𝐷𝐸𝐹, 𝑥 2 = 12, so 𝑥 =

, which means both 𝐸𝐹 and 𝐷𝐹 are

units long.

Unit 5 | 431

Practice Problems 1. Find the lengths of the legs. A. 4 2 units 4

units

C. 4 units

D. Not enough information

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2. What is the length of the diagonal?

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3. A square has a diagonal of length 5 cm. What is the area of the square?

4. Triangle 𝐿𝑃𝑀 is a 45 − 45 − 90 triangle where 𝑚∠𝐿𝑀𝑃 = 𝑚∠𝑀𝐿𝑃 = 45°, 𝑚∠𝐿𝑃𝑀 = 90°, and 𝐿𝑀 = 7. What are the lengths of 𝑃𝑀 and 𝐿𝑃?

432 | Unit 5

Review Problems

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5. Priya is teaching her younger cousin to ride a bike. She wants to stay on roads that are not too steep and easy enough for a new bike rider. She has decided the roads must have an angle less than or equal to 7°. A 7° angle in a right triangle has a 3 ∶ 25 ratio for the legs. List the legs of 2 right triangles that would be safe for a new bike rider.

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6. Clare and Han are discussing how to find the missing lengths. Clare says she is using similarity. Han says he is using the Pythagorean theorem.

a. Do you agree with either of them? Show or explain your reasoning.

b. Find the missing sides.

Unit 5 | 433

7. In right triangle 𝐴𝐵𝐶, ∠𝐶 is a right angle, 𝐴𝐵 is 25 units long, and 𝐵𝐶 is 24 units long. What is the length of 𝐴𝐶? A. 1 B. 2 D. 49

8. Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are shown.

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a. Find the length of 𝐸𝐹.

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C. 7

b. Find the measure of ∠𝐸.

434 | Unit 5

Unit 5, Lesson 9: Half an Equilateral Triangle Warm-Up: Notice and Wonder: Triangle Slices

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Sketch an equilateral triangle and an altitude from any vertex in the equilateral triangle.

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What do you notice? What do you wonder?

Unit 5 | 435

Guided Activity: Decomposing Equilateral Triangles

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1. An equilateral triangle is shown with side length 2 units and an altitude drawn. Find the values of 𝑥 and 𝑦.

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2. Measure several more of these “half equilateral triangles” by drawing equilateral triangles and altitudes. Compute the ratios of the side lengths of these new triangles.

Each half of an equilateral triangle forms a 30 − 60 − 90 triangle.

3. Complete the table by describing the relationship between the side lengths of a 30 − 60 − 90 triangle. Sides

Relationship

Short leg and hypotenuse

Short leg and long leg

436 | Unit 5

4. The table includes 3 different 30 − 60 − 90 triangles. ∆𝑽𝑯𝑲

∆𝑯𝑹𝒀

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∆𝑲𝑩𝑳

a. Discuss with your partner the relationships between the sides of a 30 − 60 − 90 triangle in the table.

Short Side

Work to Find the Long Side

∆𝐾𝐵𝐿

4 3

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Triangle

b. Complete the table by showing how the length of the short side of a triangle can be used to determine the lengths of the long side and the hypotenuse. Write each length as an exact value.

∆𝑉𝐻𝐾

∆𝐻𝑅𝑌

2 7

Work to Find the Hypotenuse

Unit 5 | 437

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5. Describe how the triangle shows the relationships in a 30 − 60 − 90 triangle.

Collaborative Activity: Generalize Half Equilateral Triangles

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Calculate the lengths of the 5 unlabeled sides.

𝐴𝐶 =

438 | Unit 5

𝐹𝐻 =

𝐻𝐺 =

𝑀𝑁 =

𝐿𝑀 =

Lesson Summary Drawing the altitude of an equilateral triangle decomposes the equilateral triangle into 2 congruent triangles. They are right triangles with acute angles of 30° and 60°. These congruent angles make all triangles with angles 30°, 60°, and 90° similar by angle-angle (AA) triangle similarity conditions.

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If a right triangle has angle measures of 30°, 60°, and 90° and the shortest side is 1 unit long, then the hypotenuse must be 2 units long since the triangle can be thought of as half of an equilateral triangle. If the length of the altitude is 𝑎, by the Pythagorean theorem 𝑎2 + 12 = 22, so 𝑎 = 3.

Another right triangle with angle measures of 30°, 60°, and 90°, and with the shortest side 𝑦 units long, is shown. By AA triangle similarity, it must be similar to the right triangle with angles 30°, 60°, and 90° and with sides 1, 3, and 2 units long. The scale factor is 𝑦, so a triangle with angles 30°, 60°, and 90° has side lengths 𝑦, 𝑦 3, and 2𝑦 units long.

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Triangles 𝐴𝐵𝐶 and 𝐸𝐺𝐻 are shown.

In ∆𝐴𝐵𝐶, 2𝑦 = 5, so 𝑦 = 5 . That means 𝐷𝐵 is 5 units and 𝐷𝐶 is 5 3 units. 2

In ∆𝐸𝐺𝐻, 𝑦 3 = 4, so 𝑦 =

. That means 𝐹𝐺 is

units and 𝐸𝐺 is 2

units.

Unit 5 | 439

Practice Problems

□ Angles 𝐵 and 𝐶 are 60°. □ 𝑥=3 3 □ 𝑥=6 3 □ ∆𝐴𝐵𝐷 ≅ ∆𝐴𝐶𝐷 □ 𝐵𝐷 and 𝐶𝐷 are both 3 units long.

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2. Find the length of each leg.

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1. Select all statements that are true about equilateral triangle 𝐴𝐵𝐶.

3. An equilateral triangle has a side length of 10 units. What is its area?

440 | Unit 5

4. Triangle 𝑇𝑁𝑊 is a right triangle where 𝑚∠𝑁𝑇𝑊 = 60°, 𝑚∠𝑊𝑁𝑇 = 90°, and 𝑇𝑊 = 18. What are the measures of the other sides of the triangle?

Review Problems

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5. Find the lengths of the legs.

6. A square has side length 3 units. What is the length of the diagonal? 3

units

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C. 3 2 units

A. 3 units

D. 6 units

7. A step has a height of 5 inches (in.). A ramp starts 4 feet (ft.) away from the base of the step, making a 5.9° angle with the ground. What can you say about the angle the ramp would make with the ground if the ramp starts farther away from the step?

A. The angle would decrease.

B. The angle would remain the same. C. The angle would increase. D. We cannot determine anything about the angle.

Unit 5 | 441

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8. Segment 𝐴′𝐵′ is parallel to 𝐴𝐵.

a. What is the length of 𝐴′𝐵′?

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b. What is the length of 𝐵′𝐵?

442 | Unit 5

Unit 5, Lesson 10: Ratios in Right Triangles

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Warm-Up: Ratio Rivalry

Consider 𝑎 and 𝑏 . Which is greater, or are they equal? Explain how you know. 𝑑

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𝑐

Unit 5 | 443

Exploration Activity: Tons of Triangles

This activity requires the use of an applet, so please make your way over to the digital platform to find the link.

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Your teacher will give you some angles.

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1. Use the applet to build 4 different right triangles for each of your angles.

2. Record the side lengths of each of the triangles.

3. Compute these 3 quotients for the acute angles in each triangle. a. The length of the leg adjacent to your angle divided by the length of the hypotenuse.

444 | Unit 5

b. The length of the leg opposite from your angle divided by the length of the hypotenuse.

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c. The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle.

4. Find the mean of each type of quotient.

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5. What do you notice? What do you wonder?

Unit 5 | 445

Collaborative Activity: Tons of Ratios

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1. Compare the row for 20° and the row for 70° in the right triangle table. What is the same? What is different?

2. The row for 55° is given here. Complete the row for 35°.

0.574

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55°

Opposite leg ÷ hypotenuse

35°

Adjacent leg ÷ hypotenuse

Angle

0.819

Opposite leg ÷ adjacent leg

1.428

3. What do you know about a triangle with an adjacent leg to hypotenuse ratio value of 0.839?

446 | Unit 5

Lesson Summary All right triangles that contain the same acute angles are similar to each other. This means that the ratios of corresponding side lengths are equal for all right triangles with the same acute angles.

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For example, the 3 triangles shown are all similar by angle-angle (AA) triangle similarity conditions.

By focusing on the 25° angles, it can be shown that the ratio of the adjacent leg to the hypotenuse for all 3 triangles is approximately 0.91.

𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐥𝐞𝐠 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠 𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐥𝐞𝐠

0.819

0.574

0.700

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Angle

Because all right triangles with the same acute angle measures have the same ratios of side lengths, patterns in these relationships can be used to solve problems. The right triangle table shown was created by finding ratios of side lengths in several right triangles with different angle measures.

25∘ 35∘ 45∘

55∘ 65∘

0.906 0.707 0.574 0.423

0.423 0.707 0.819 0.906

0.466 1.000 1.428 2.145

Some ratios in this table are repeated. Notice that the rows for 25° and 65° have 2 of the same ratios. What is special about 25° and 65°? They are complementary angles, which means the 2 angles sum to 90°. This seems to be true for other complementary angles. Notice that 35 + 55 = 90, and those rows both have 0.819 as a ratio. These relationships will be explored more deeply in upcoming lessons. © Accelerate Learning Inc. - All Rights Reserved

Unit 5 | 447

Practice Problems 1. Angle 𝐵 is an acute angle in a right triangle. What is a reasonable approximation for ∠𝐵 if the ratio for the opposite leg divided by the hypotenuse is 0.67?

0.97

0.26

Opposite leg ÷ adjacent leg

0.27

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𝐶

Opposite leg ÷ hypotenuse

𝐴

Adjacent leg ÷ hypotenuse

Angle

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2. Estimate the values to complete the table.

3. Priya says, “I know everything about a right triangle with a 30° angle and a hypotenuse with length 1 centimeter (cm). Here, look.“ • The other angle is 60°.

• The leg adjacent to the 30° angle is 0.866 cm long. • The side opposite the 30° angle is 0.5 cm long.

Han asks, “What would happen if a right triangle with a 30° angle has a hypotenuse that is 2 cm instead?“

Help them find the missing angles and side lengths in the new triangle. Explain or show your reasoning.

448 | Unit 5

Review Problems 4. Triangle 𝐴𝐵𝐶 is equilateral.

b. What is the measure of ∠𝐵?

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a. What is the value of 𝑥?

5. An equilateral triangle has side length 8 units. What is the area?

C. 24 3 square units

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D. 32 square units

B. 24 square units

A. 16 3 square units

6. What is the length of the square’s side?

A. 3 units B.

units

C. 6 2 units D. 12 units

Unit 5 | 449

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Unit 5, Lesson 11: Working with Ratios in Right Triangles Warm-Up: Launch Pad

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When a rocket is launched, it climbs 50 feet (ft.) for every 13 ft. it travels horizontally. Draw a diagram to represent the situation. Then estimate the rocket’s launch angle.

Unit 5 | 451

Exploration Activity: Pythagorean Triples

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1. Sketch the triangle with side lengths 7, 24, and 25 units. Label the smallest ∠𝐴.

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2. Find the 3 ratios of side lengths for ∠𝐴.

3. Estimate the acute angles in this triangle.

452 | Unit 5

Collaborative Activity: Solve All the Triangles

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1. What is the length of 𝐴𝐵?

2. In a right triangle with one angle measuring 40°, the leg opposite the 40° angle is 5 centimeters (cm). What is the length of the hypotenuse?

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3. What is the length of segment 𝐷𝐸?

4. In a right triangle with one angle measuring 70°, the leg opposite the 70° angle is 12 cm. What is the length of the leg adjacent to the 70° angle?

Unit 5 | 453

Lesson Summary All right triangles that contain the same acute angle are similar. This means that the ratios of corresponding side lengths are equal for all right triangles with the same acute angles. Using the ratios calculated in the previous lesson and properties of similar triangles, unknown side lengths and angles in right triangles can be calculated and estimated.

𝑏

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If the legs of any right triangle with an angle of 25° are measured, the ratio of the leg opposite the 25° angle to the leg adjacent to the 25° angle will always be 0.466. Therefore, length 𝑏 can be found. Since 5 = 0.466, 𝑏 is 10.7 units.

Similarly, the measures of the missing angles in ∆𝐷𝐸𝐹 can be estimated as shown in the table. 𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐥𝐞𝐠 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠 𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐥𝐞𝐠

0.866

1.732

Angle

0.643

50∘

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60∘

0.500

0.766

1.192

• The ratio of the leg opposite ∠𝐷 to the hypotenuse is 0.794. This value is between the value of the opposite leg divided by the hypotenuse for 50° (0.766) and 60 degrees (0.866), so the measure of ∠𝐷 must be between 50° and 60°. • Similarly, the leg opposite ∠𝐷 divided by the leg adjacent to ∠𝐷 gives a ratio of 1.283, which is between the same ratio for 50° (1.192) and 60° (1.732).

• The exact value turns out to be 52°.

454 | Unit 5

Practice Problems 1. A triangle has sides with lengths 8, 15, and 17.

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a. Verify this is a Pythagorean triple.

b. Approximate the acute angles in this triangle.

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2. Kiran is flying a kite. He gets tired, so he stakes the kite into the ground. The kite is on a string that is 18 ft. long and makes a 30° angle with the ground. How high is the kite?

3. Triangle 𝐴𝐵𝐶 has a right angle at 𝐶. Select all measurements which would mean it has a hypotenuse with a length of 10 units.

□ 𝑚∠𝐴 = 20° and 𝐵𝐶 = 2 □ 𝐴𝐶 = 7 and 𝐵𝐶 = 3 □ 𝑚∠𝐵 = 50° and 𝐵𝐶 = 4 □ 𝑚∠𝐴 = 30° and 𝐵𝐶 = 5 □ 𝐴𝐶 = 8 and 𝐵𝐶 = 6

Unit 5 | 455

Review Problems 4. What is a reasonable approximation for ∠𝐵 if the ratio of the adjacent leg divided by the hypotenuse is 0.45? A. 27° B. 30°

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C. 60° D. 63°

Adjacent leg ÷ hypotenuse

𝐴

0.31

Opposite leg ÷ hypotenuse

Opposite leg ÷ adjacent leg

0.95

3.1

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Angle

5. Estimate the values to complete the table.

𝐶

6. Find the lengths of 𝐴𝐷 and 𝐵𝐷. Then check your answers using a different method.

456 | Unit 5

Unit 5, Lesson 12: Working with Trigonometric Ratios Warm-Up: This Time with Strategies

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Estimate the value of 𝑧.

Unit 5 | 457

Exploration Activity: New Names, Same Ratios 1. Use your calculator to determine the values of cos(50), sin(50), and tan(50). cos(50) = sin(50) =

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tan(50) =

2. Use your calculator to determine the values of cos(40), sin(40), and tan(40). cos(40) = tan(40) =

sin(40) =

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3. How do these values compare to your chart?

4. Find the value of 𝑧.

458 | Unit 5

Collaborative Activity: Using Trigonometric Ratios

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1. Triangle 𝐵𝑊𝐻 is shown, where ℎ = 12.8, 𝑤 = 8, 𝑥 = 51.3°, and 𝑦 = 38.7°.

a. Select all the equations that can be used to find the value of 𝑏.

□ cos(51.3°) = 8𝑏

𝑏 □ sin(38.7°) = 12.8

𝑏 □ tan(51.3°) = 12.8

□ tan(38.7°) = 8𝑏

𝑏 □ cos(38.7°) = 12.8

𝑏 □ sin(51.3°) = 12.8

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b. Determine the value of 𝑏. Round to the nearest thousandth.

2. Triangle 𝑆𝐺𝐾 is shown.

a. Use a trigonometric ratio for 𝜃 to write an equation that can be solved for the value of 𝑘.

b. Which trigonometric ratio for 𝜃 cannot be used to write an equation that can solved for the value of 𝑘?

Unit 5 | 459

3. Triangle 𝑃𝑀𝐻 is shown. Work with your partner to determine which trigonometric ratio can be used to determine the unknown side or angle. Then, use that ratio to write an equation that can be solved for the unknown value. In writing your equations, use the variables ℎ, 𝑝, 𝑚, 𝑥, and 𝑦.

𝑚∠𝑃𝐻𝑀 and length of 𝑃𝑀

𝑃𝐻

𝑚∠𝑃𝐻𝑀 and length of 𝑃𝑀

𝑀𝐻

𝑚∠𝐻𝑃𝑀 and length of 𝑃𝑀

𝑃𝐻

Lengths of 𝑀𝐻 and 𝑃𝐻

𝑚∠𝐻𝑃𝑀

Equation

sine cosine tangent sine cosine tangent sine cosine tangent sine cosine tangent sine cosine tangent

cr 𝑚∠𝑃𝐻𝑀

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Lengths of 𝑀𝐻 and 𝑃𝑀

Trigonometric Ratio

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Unknown

Given

Guided Activity: Solve These Triangles

1. Solve for 𝑥.

460 | Unit 5

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2. Solve for 𝑦.

3. Find all the missing sides and angle measures.

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a. The measure of ∠𝑋 is 90° and ∠𝑌 is 12°. Side 𝑋𝑍 has length 2 centimeters (cm).

Unit 5 | 461

Lesson Summary The ratio of adjacent leg is used so frequently that it has a name. It is called the cosine of hyptoenuse the angle.

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The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse. It is written as cos(25) to say “the cosine of 25°.“

A scientific calculator can display the cosine of any angle, which means you can more precisely calculate unknown side lengths rather than estimating using the table. The right triangle table is sometimes called a trigonometry table because cosine, sine, and tangent are trigonometric ratios.

The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse.

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The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg. The table shows each ratio for a 25° angle in a right triangle with their special names. Angle 25∘

cos(𝜽) = 𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐥𝐞𝐠 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

cos(25) = 0.906

𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

sin(25) = 0.423

tan(𝜽) = 𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠 𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐥𝐞𝐠

tan(25) = 0.466

Consider the triangle shown.

sin(𝜽) = 𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠

If the length of 𝑏 is 7, the length of 𝑐 can be found by solving the equation cos(25) = 7 . Therefore, 𝑐 is about 𝑐 7.7 units.

The Pythagorean theorem, sine, or tangent can be used to solve for 𝑎. Using tangent, tan(25) = 𝑎 , so 𝑎 is about 3.3 units. 7

The answers can be checked using the Pythagorean theorem. It should be true that 3.32 + 72 = 7.72. The expressions are almost equal, which makes sense because we expect some error due to rounding. 462 | Unit 5

Practice Problems 1. Select all the true statements. 4

□ sin(𝜃) = 97 □ tan(𝛽) = 94

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□ tan(𝛽) = 49

□ cos(𝛽) = 97 □ 42 + 92 = 97

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2. Write an expression that can be used to find the length of 𝐽𝐻 and an expression that can be used to find the length of 𝐺𝐽 𝐺.

3. Andre and Clare are discussing ∆𝐴𝐵𝐶 that has a right angle at 𝐶 and a hypotenuse of length 15 units. Andre thinks the triangle could have legs that are 9 and 12 units long. Clare thinks ∠𝐵 could be 20° and then side 𝐵𝐶 would be 14.1 units long. Do you agree with either of them? Explain or show your reasoning.

Unit 5 | 463

4. In ∆𝑍𝑀𝐸, side 𝑀𝐸 = 14.9 and 𝑚∠𝑍𝐸𝑀 = 41.5°.

a. If ∠𝑍𝑀𝐸 is a right angle, what is the length of 𝑍𝑀?

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b. Sharif used the information to determine the length of 𝑍𝑀 given that ∠𝑍𝑀𝐸 is a right angle. Her work is shown below. 14.9

tan 41.5° = 𝑍𝑀

𝑍𝑀 ∙ tan 41.5° = 14.9 14.9

𝑍𝑀 = tan 41.5° 𝑍𝑀 ≈ 16.841

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Find Sharif’s error and correct it.

Review Problems

5. A triangle has sides with lengths 5, 12, and 13.

a. Verify this is a Pythagorean triple.

b. Approximate the acute angles in this triangle.

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6. Lin missed class and Tyler is helping her use the table to approximate the angle measures that have the ratios listed. Tyler says, “You can use the right triangle table to figure this out.” Lin notices that some of the ratios are the same in each row. Estimate the angles and explain why some of the values are repeated. Angle

Adjacent leg ÷ hypotenuse

Opposite leg ÷ hypotenuse

Opposite leg ÷ adjacent leg

0.819

0.573

0.700

0.819

1.428

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0.573

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7. Find the length of each leg.

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Unit 5, Lesson 13: Applying Ratios in Right Triangles Warm-Up: Tilted Triangle

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Calculate the lengths of sides 𝐴𝐶 and 𝐵𝐶.

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Collaborative Activity: Info Gap: Trigonometry Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the data card:

If your teacher gives you the problem card:

2. Ask your partner, “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)

2. Ask your partner for the specific information that you need.

3. Explain to your partner how you are using the information to solve the problem. 4. When you have enough information, share the problem card with your partner, and solve the problem independently.

3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”

1. Silently read your card and think about what information you need to answer the question.

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1. Silently read the information on your card.

4. Read the problem card, and solve the problem independently.

5. Read the data card, and discuss your reasoning.

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5. Share the data card, and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

468 | Unit 5

Guided Activity: Tallest Tower 1. The tallest building in the world is the Burj Khalifa in Dubai (as of April 2019).

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If you’re standing on the bridge 250 meters (m) from the bottom of the building, you have to look up at a 73° angle to see the top. How tall is the building? Explain or show your reasoning.

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2. The tallest masonry building in the world is City Hall in Philadelphia (as of April 2019). If you’re standing on the street 1,300 feet (ft.) from the bottom of the building, you have to look up at a 23° angle to see the top. How tall is the building? Explain or show your reasoning.

Unit 5 | 469

Lesson Summary

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Measures in different right triangles can be calculated and estimated using trigonometry and properties of right triangles. These skills are used to estimate unknown heights of objects that are too tall to measure directly. For example, the top of this tree can’t be reached with a measuring tape.

To calculate the height of the tree, stand where the angle between the top and bottom of the tree is 10°. Since you know the distance to the tree (the adjacent leg) and would like to know the height (the opposite leg), use tangent. For the triangle created, tan(10) = ℎ . In the calculator, it shows that tan(10) ≈ 0.176, which can be used to 100

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calculate that ℎ is about 17.6. That means the tree is approximately 17.6 ft. tall.

Practice Problems

1. Technology required. Mai is visiting Paris to see the Eiffel Tower. She is 80 ft. away when she spots it. To see the top, she has to look up at an angle of 85.7°. How tall is the Eiffel Tower?

470 | Unit 5

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2. Technology required. Find the missing measurements of the right triangle.

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Review Problems

3. Technology required. Gateway Arch in St. Louis, Missouri, is 630 ft. tall. Priya can look up at a 50° angle to see the top of the arch. How far away from the base of the arch is she standing?

4. Based on the figure, which equation is true?

A. sin(32) = 8.5 𝑥

B. sin(32) = 𝑥

8.5

C. cos(32) = 8.5 𝑥

D. cos(32) = 𝑥

8.5

Unit 5 | 471

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5. Kiran is flying a kite. He gets tired, so he stakes the kite into the ground. The kite is on a string that is 12 ft. long and makes a 45° angle with the ground. How high is the kite?

A. 12 ft. B.

ft.

D. 24 ft.

C. 12 2 ft.

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6. In the right triangles shown, the measure of ∠𝐴𝐵𝐶 is the same as the measure of ∠𝐸𝐵𝐷. What is the length of side 𝐵𝐸?

472 | Unit 5

Unit 5, Lesson 14: Sine and Cosine in the Same Right Triangle Warm-Up: Which One Doesn’t Belong: Four Triangles Which one doesn’t belong? B.

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Exploration Activity: Twin Triangles 1. Your teacher will assign you to either Column A or Column B. Find the value of the variable for the problems in your column. Column B:

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Column A:

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2. Compare your solutions with your group’s solutions. Why did you get the same answers to different problems?

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Collaborative Activity: Explain the Connection

2. Explain why sin(𝜃) = cos(90 − 𝜃).

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1. Draw a diagram that will help you explain why sin(𝜃) = cos(90 − 𝜃).

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3. Discuss your thinking with your group. If you disagree, work to reach an agreement. Create a visual display that includes: • A clearly labeled diagram.

• An explanation using precise language.

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Lesson Summary

25∘ 65∘

𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐥𝐞𝐠

𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠

cosine(𝜽) = 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

sine(𝜽) = 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

0.423

0.906

0.423

Angle

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In previous lessons you learned that any right triangle with acute angles of 25° and 65° was similar to any other right triangle with these same acute angles. Revisiting these triangles, notice that the sine of 25° is equal to the cosine of 65°, and the cosine of 25° is equal to the sine of 65°.

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Looking at a general right triangle, the angles can be written as 90°, 𝜃°, and (90 − 𝜃)°. Mathematicians often use Greek letters to represent angles. For instance, 𝜃 is a Greek letter used frequently in trigonometry.

Angle 𝜃∘

(90 − 𝜃)∘ 476 | Unit 5

cosine(𝜽) = 𝑥 ℎ

𝑦 ℎ

𝐚𝐝𝐣𝐚𝐜𝐞𝐧𝐭 𝐥𝐞𝐠 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

sine(𝜽) =

𝑦 ℎ

𝐨𝐩𝐩𝐨𝐬𝐢𝐭𝐞 𝐥𝐞𝐠 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

𝑥 ℎ

Practice Problems 1. Select all the true equations.

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□ cos(15) = sin(15) □ cos(75) = sin(15) □ cos(75) = cos(15) □ cos(15) = sin(75) □ tan(15) = tan(75)

2. Write 2 expressions that can be used to find the value of 𝑥.

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3. Andre and Mai are discussing how to solve for side 𝐴𝐵. Andre thinks he can use the equation tan(12) = 𝑥 to solve for 𝐴𝐵. Mai thinks she can use the equation tan(78) = 6 6

𝑥

to solve for 𝐴𝐵. Do you agree with either of them? Show or explain your reasoning.

Unit 5 | 477

Review Problems

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4. Technology required. Jada is visiting New York City to see the Empire State building. She is 100 feet away when she spots it. To see the top, she has to look up at an angle of 86.1°. How tall is the Empire State building?

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5. Technology required. Find the missing measurements in ∆𝐴𝐵𝐶.

6. Right triangle 𝐴𝐵𝐶 is shown.

Write 2 expressions which are equal to the length of side 𝐵𝐶.

478 | Unit 5

Unit 5, Lesson 15: Using Trigonometric Ratios to Find Angles Warm-Up: Once More with the Table

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A triangle with side lengths 3, 4, and 5 is a right triangle by the converse of the Pythagorean theorem. What are the measures of the acute angles?

Guided Activity: Using Inverse Trigonometric Functions 1. In Δ𝐻𝑅𝐶, 𝑚∠𝐻𝑅𝐶 = 90°, 𝑅𝐶 = 8, and 𝐻𝐶 = 13.

The equation cos(𝜃) = 8 can be used to find 𝜃. To find 𝜃 using a calculator, enter

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13 8 cos � � Round any angle measure below to the nearest whole degree. 13 a. What is cos−1 � 8 �? 13 −1

b. Write an equation that can be used to find 𝑚∠𝐶𝐻𝑅.

c. Find 𝑚∠𝐶𝐻𝑅 using an inverse trigonometric function on a calculator. Be sure the calculator is in degree mode.

Unit 5 | 479

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2. Triangle 𝑊𝐿𝑇 is a right triangle where 𝑚∠𝑊𝑇𝐿 = 90°, 𝑇𝐿 = 11, and 𝑊𝐿 = 23. What is 𝑚∠𝑇𝑊𝐿? Round to the nearest whole degree.

3. Triangle 𝑊𝐵𝑁 is shown, where ∠𝐵 is a right angle.

a. Determine the measure of ∠𝐵𝑁𝑊 if 𝑛 = 36 and 𝑤 = 77.

b. Gabriella used sin−1 � 77 �to find the measure of ∠𝐵𝑁𝑊.

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Determine whether sin−1� 77 � results in the same measure 85 as your answer.

c. Discuss with your partner whether Gabriella’s method is valid. Write a justification of your decision.

480 | Unit 5

Collaborative Activity: From Ratios to Angles

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Find all missing side and angle measures.

Unit 5 | 481

Collaborative Activity: Leaning Ladders A good rule of thumb for a safe angle to use when leaning a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground.

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1. What are the angles in the triangle formed by your body and the ladder?

2. What are the angles in the triangle formed by the ladder, the ground, and the railing? Explain or show your reasoning.

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3. You have a 13 foot (ft.) long ladder and need to climb to a 12 ft. tall roof. a. If you put the top of the ladder at the top of the wall, what angle is formed between the ladder and the ground?

b. Is it possible to adjust the ladder to a safe angle? If so, give someone instructions to do so. If not, explain why not.

482 | Unit 5

Lesson Summary The missing sides and angles of right triangles can be found using the trigonometric ratios and a calculator. The right triangle table can be used to estimate angle measures as in previous lessons. However, with a calculator, you can find angles more precisely using inverse trigonometric functions. 𝑐

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• The inverse of cosine, cos−1 � 𝑎 �, is arccosine. • The inverse of sine, sin−1� 𝑏 �, is arcsine. 𝑐

• The inverse of tangent, tan−1 � 𝑏 �, is arctangent. 𝑎

The arccosine of a number between 0 and 1 is the acute angle whose cosine is that number.

The arcsine of a number between 0 and 1 is the acute angle whose sine is that number.

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The arctangent of a positive number is the acute angle whose tangent is that number. These functions can be used to determine the measure of an angle, where 𝑎 is the side adjacent to the angle, 𝑏 is the side opposite to the angle, and 𝑐 is the hypotenuse. Triangle 𝐴𝐵𝐶 is shown.

The side opposite ∠𝐴 is 3 units long, and the side adjacent to ∠𝐴 is 12 units long. Therefore, to find ∠𝐴, write an equation using tangent: tan(𝛼) = 3 12

To find the measure of ∠𝐴, the calculator can be used to answer the question, “What angle has a tangent of 3 ?” To answer that, use 12 arctangent by writing tan−1� 3 �. Because 𝛼 = tan−1� 3 �, ∠𝐴 measures about 14°. If the 12

cosine is known, use arccosine to look up the angle, and if the sine is known, use arcsine. Angle 𝐵 can be calculated using another trigonometric equation or triangle angle sum

theorem. If arctangent is used again and tan(𝜃) = 12 , then 𝜃 = tan−1� 12 �, which is about 3

76°. This matches the answer using the triangle angle sum theorem: 180 − 90 − 14 = 76.

Unit 5 | 483

Practice Problems

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1. Technology required. Ramps in a parking garage need to be both steep and safe. The maximum safe incline for a ramp is 8.5°. Is this ramp safe? If not, provide dimensions that would make the ramp safe.

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2. Technology required. 𝐴𝐵𝐶𝐷 is a rectangle. Find the length of 𝐴𝐶 and the measures of ∝ and 𝜃.

3. Technology required. Find the missing measurements.

484 | Unit 5

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4. Triangle 𝐾𝑁𝑃 is a right triangle where 𝐾𝑃 = 46 and 𝐾𝑁 = 27.3. If ∠𝑃𝑁𝐾 is a right angle, what is 𝑚∠𝐾𝑃𝑁?

Review Problems

B. 3.5 cos(17) C. 3.5 sin(17) D. sin(17)

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3.5

A. 3.5 tan(17)

5. A rope with a length of 3.5 meters is tied from a stake in the ground to the top of a tent. It forms a 17° angle with the ground. How tall is the tent?

6. Technology required. What is the value of 𝑥?

Unit 5 | 485

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7. Find the missing side in each triangle using any method. Check your answers using a different method.

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8. The triangles are congruent. Write a sequence of rigid motions that takes ∆𝑋𝑌𝑍 onto ∆𝐵𝐶𝐴.

486 | Unit 5

Unit 5, Lesson 16: Solving Problems with Trigonometry Warm-Up: Notice and Wonder

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1. The picture shows the stairs of the Hunting Island Lighthouse in South Carolina.

a. What do you notice?

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b. What do you wonder?

c. The lighthouse has 167 steps. Do you think this picture includes all of the steps? Explain your answer.

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Exploration Activity: Angles of Elevation and Depression Real-world trigonometric ratio problems involve angles that are made from a horizontal line of sight.

2. Complete the statement. The horizontal line of sight is always

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1. Use a ruler to draw the man’s horizontal line of sight.

perpendicular parallel

to the ground.

3. The angles created from the horizontal line of sight are called angles of elevation and angles of depression.

In the image, Hector is looking at birds in the sky and a dog on the ground.

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Label the angle of elevation as 𝐸. Label the angle of depression as 𝐷.

4. Based on the name of the angle, work with your partner to write a definition for an angle of elevation.

5. Work with your partner to write a definition for an angle of depression.

6. Compare your definitions with another group’s. Discuss any differences. Make changes to your definitions if necessary. 488 | Unit 5

Guided Activity: Angles of Elevation and Depression

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1. The angle of depression, 𝑚∠𝐷𝑊𝑀, from the airplane to the runway, point 𝑀, measures 37°.

a. Explain why ∠𝐷𝑊𝑀 is congruent to ∠𝑅𝑀𝑊.

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b. If the airplane is flying at an altitude of 2,500 feet, 𝑊𝑅, and is descending to the airport, point 𝑀, what is the distance, 𝑀𝑊, that the plane has to travel?

c. How far is the runway from the point on the ground directly below the airplane, point 𝑅?

Unit 5 | 489

2. Kamala is visiting the Hunting Island Lighthouse on Saint Helena Island, South Carolina. Kamala is standing at point 𝐾 and looking at the top of the lighthouse, point 𝐿. Line segment 𝐾𝑇 is at Kamala’s eye level, which is 1.75 meters (m) high.

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a. When Kamala looks at the top of the lighthouse, is the angle created an angle of elevation or an angle of depression?

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b. The height of the Hunting Island Lighthouse is 41.45 m. Kamala is standing 15.25 m from the lighthouse. Label ∆𝐾𝑇𝐿 with this information.

c. What is the measure of the angle created when Kamala looks at the top of the lighthouse?

490 | Unit 5

Collaborative Activity: Trigonometry in the Real World

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1. A cell tower was built in 2 sections, with a control shed built 60 feet (ft.) from the base of the tower on level ground. From the base of the control shed, the angle of elevation, ∠𝐿𝑆𝑊, is 22° to the top of the first section of the tower, point 𝑊. To the top of the second section, the angle of elevation, ∠𝐿𝑆𝑅, is 46°.

a. Label the diagram with the given information.

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b. Determine the length of 𝑅𝑊, the second section of tower, in ft. Round to the nearest thousandth of a foot.

2. A hot air balloon is flying 38.5 ft. above one end of a soccer field. The soccer field is 200 ft. long and is directly below the balloon.

a. Draw and label a diagram illustrating the given information.

b. What is the angle of elevation from the other end of the soccer field to the hot air balloon?

c. The hot air balloon rises higher into the air. If the new angle of elevation from the other end of the soccer field to the balloon is 17.3°, how many ft. did the hot air balloon rise? © Accelerate Learning Inc. - All Rights Reserved

Unit 5 | 491

Lesson Summary

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Trigonometric ratios and the Pythagorean theorem can be used to calculate the missing sides and angles of right triangles. These strategies can also be used to solve right triangle problems in real-world situations. In addition to other examples explored throughout the unit, real-world trigonometric problems can include angles that are made from a horizontal line of sight. The horizontal line of sight is always parallel to the ground. The angles created from the horizontal line of sight are called angles of elevation and angles of depression. • An angle of depression is the angle formed by a horizontal line and the downward line of sight to an object below. • An angle of elevation is the angle formed by a horizontal line and the upward line of sight to an object above.

For example, suppose a student is standing away from a tree and is looking at a monkey on top of the tree. Triangle 𝐿𝑀𝑇 is shown to represent this situation.

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Angle 𝑇𝐿𝑀 is an angle of elevation because the angle is formed by the student’s line of sight, 𝐿𝑇, and the side length 𝐿𝑀 above the line of sight.

492 | Unit 5

Practice Problems 1. A swimming pool is 48 ft. long and 4 ft. deep in the shallow end. There is a steady decline of 14° toward the deep end of the pool. Rectangle 𝑅𝐿𝑆𝑌 and ∆𝑆𝑌𝐸 are shown to model the transition from the shallow end to the deep end. Label the diagram with the given information.

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a. How deep is the deepest point of the pool?

b. Line segment 𝐸𝑆 represents the slant in the pool from the shallow end to the deep end. What is the length of 𝐸𝑆?

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2. Technology required. The sun is 62° above the horizon. A tree casts a shadow that is 12 ft. long. How tall is the tree?

Unit 5 | 493

3. Technology required. A plane leaves the ground with an elevation angle of 6°. The plane travels 10 miles horizontally. a. How high is the plane at the time?

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b. What is the distance of the plane’s path?

Review Problems

□ cos (37) = sin (53) □ tan (37) = tan (53) □ sin (37) = cos (53) □ sin (37) = sin (53) □ cos (𝜃) = sin (90 − 𝜃)

4. Select all true equations.

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5. Technology required. Clare is flying a kite. She gets tired, so she stakes the kite into the ground. The kite is on a string that is 30 ft. long and makes a 27° angle with the ground. How high is the kite? A. 30 ft.

B. 13.6 ft. C. 26.7 ft.

D. 15.3 ft.

6. Technology required. Find the missing measurements.

494 | Unit 5

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