Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass.
1 Scientific Notation, Exponents, and Irrational Numbers
2 Rigid Motions and Congruent Figures
3 Dilations and Similar Figures
4 Linear Equations in One and Two Variables
5 Systems of Linear Equations
6 Functions and Bivariate Statistics
Student Edition: Grade 8, Module 2, Contents
Rigid Motions and Congruent Figures
Topic A 5
Rigid Motions and Their Properties
Lesson 1
Motions of the Plane
Lesson 2
Translations
Lesson 3
Reflections
Lesson 4
Translations and Reflections on the Coordinate Plane
Lesson 5
Rotations
Lesson 6
Rotations on the Coordinate Plane
Topic B
Rigid Motions and Congruent Figures
Lesson 7
Working Backward
Lesson 8
Sequencing the Rigid Motions
Lesson 9
Ordering Sequences of Rigid Motions
Lesson 10
Congruent Figures
Lesson 11
Showing Figures Are Congruent
23
39
53
Topic C
Angle Relationships
Lesson 12
Lines Cut by a Transversal
Lesson 13
Angle Sum of a Triangle
Lesson 14
Showing Lines Are Parallel
Lesson 15
Exterior Angles of Triangles
Lesson 16
69
83
103
105
Find Unknown Angle Measures
Topic D
Congruent Figures and the Pythagorean Theorem
Lesson 17
Proving the Pythagorean Theorem
Lesson 18
Proving the Converse of the Pythagorean Theorem
Lesson 19
Using the Pythagorean Theorem and Its Converse
Lesson 20
Distance in the Coordinate Plane
Lesson 21
Applying the Pythagorean Theorem
Lesson 22
On the Right Path
Resources
Fluency Resources
Lesson 3 Translation
Lesson 4 Coordinate Plane
Lesson 7 Rigid Motions
Lesson 8 Rigid Motions on a Coordinate Plane
Lesson 9 Sequence of Rigid Motions
Sprint: Square Roots
TOPIC A Rigid Motions and Their Properties
Student Edition: Grade 8, Module 2, Topic A
Popular Dance Moves at “Club Geometry”
We see a dancer in one spot. Later, we see that same dancer in another spot. What happened? Well, a motion of sorts. That is motion, not in quite the usual sense but in the mathematical sense of rigid motion that we’ll explore in the coming topic. It’s a sense that includes all the above and more.
Student Edition: Grade 8, Module 2, Topic A, Lesson 1
Name Date
Motions of the Plane
1. Study the pattern.
a. Use any of the given tools and only figure A to create the pattern.
b. What tools did you use?
c. What strategy did you use?
Moving the Transparency
2. For each pair of figures, how do you move the transparency so that a traced version of the first figure lies on top of the second figure?
a. Figure A onto figure B
b. Figure A onto figure C
c. Figure A onto figure D
3. Fill in the blank to complete each sentence with one of the rigid motions: translation, reflection, or rotation.
a. I used a to map figure A onto figure B.
b. I used a to map figure A onto figure C
c. I used a to map figure A onto figure D.
Getting Technical
4. The diagram shows a figure and its image under a rigid motion.
a. What type of rigid motion occurred?
b. Can you tell which figure is the image and which is the original?
c. How would you describe the rigid motion if the figure on the left is the original? Be as specific as possible.
d. How would you describe the rigid motion if the figure on the right is the original? Be as specific as possible.
e. Measure and label every side length of the original figure and its image in centimeters.
f. Measure and label every angle within the original figure and its image.
For problems 5–9, fill in the blank to complete the sentence.
5. Translations, reflections, and rotations are all types of .
6. Rigid motions are the result of any movement of the plane in which the between any two points stays the .
7. Rigid motions map segments to segments. Rigid motions keep segment lengths the .
8. Rigid motions map angles to . Rigid motions keep angle measures the
9. Rigid motions parallel lines to lines.
Which Rigid Motion?
For problems 10–13, identify the rigid motion that maps the figure onto the image provided. Then label the vertices of the image and any known segment lengths and angle measures.
10.
Student Edition: Grade 8, Module 2, Topic A, Lesson 1
Identify the rigid motion that maps the figure onto the image provided. Then label the vertices of the image and any known segment lengths and angle measures.
Student Edition: Grade 8, Module 2, Topic A, Lesson 1
Name Date
Motions of the Plane
In this lesson, we
• defined rigid motions of the plane.
• used a transparency to identify rigid motions.
• showed that the distance between two points stays the same under rigid motions.
• labeled vertices and any known measurements of an image under a rigid motion.
Examples
Terminology
A rigid motion is the result of any movement of the plane in which the distance between any two points stays the same.
Identify the rigid motion that maps the figure onto the image provided. Then label the vertices of the image and any known segment lengths and angle measures.
Trace △ LMN onto a transparency. Move, turn, or flip the transparency so the traced version of △ LMN lies on top of its image.
Flip the transparency.
Rotation
Turn the transparency.
Translation
Move the transparency up and to the right.
A rotation maps point L to point L′ . Point L′ is the image of point L. Read the label L′ as “L prime.”
The distance between any two points stays the same under rigid motions, so segment lengths and angle measures stay the same.
Student Edition: Grade 8, Module 2, Topic A, Lesson 1
Name Date
For problems 1–3, identify the rigid motion that maps △ ABC onto its image.
4. Complete the table by identifying the rigid motion that maps the first figure onto the second figure.
1. A
Figure A
Figure B
Figure C
Figure A onto Figure B
Figure A onto Figure C
Figure B onto Figure C
Rigid Motion
For problems 5–10, identify the rigid motion that maps the figure onto the image provided. Then label the vertices of the image and any known segment lengths and angle measures.
For problems 11–13, determine whether each diagram shows a figure and its image under a rigid motion. Explain.
14. Kabir says that the two given triangles show a rigid motion because the corresponding angles have the same angle measures. Is Kabir correct? Explain.
15. Figure P′Q′R′S′ represents the image of figure PQRS under a rigid motion.
a. Label each vertex of figure PQRS.
b. Label all unknown angle measures of figure PQRS.
c. Sides P′S′ and Q′R′ are parallel. What does that tell us about the sides in figure PQRS ?
d. Figure P′Q′R′S′ is a trapezoid. Is figure PQRS also a trapezoid? Explain.
Remember
For problems 16–19, evaluate.
16. 16 + (−12) 17. 16 − (−12)
18. 16(−12) 19. 16 ÷ (−12)
20. Consider the number 0.0007.
a. Write the number in fraction form.
b. Write the number in scientific notation.
21. Rectangle ABCD has a perimeter of 32 units and an area of 48 square units.
a. If the coordinates of point A are (−8, 2), what could be the coordinates of the other three vertices? Use the coordinate plane if needed.
b. Explain how you determined the coordinates of the other three vertices.
Student Edition: Grade 8, Module 2, Topic A, Lesson 2
Name Date
Translations
1. Sketch the image of the figure based on your partner’s directions.
Direction and Distance
2. Sketch the image of the figure based on your partner’s directions.
3. Fill in the blanks to complete the sentences.
A vector is a directed line segment. Two vectors are shown.
The direction of ⟶ AB is determined by starting at point , moving along the segment, and ending at point . This direction is shown by an arrowhead placed at point . The length of a vector is the length of its underlying segment. Translate
4. Draw and label the image of point P under a translation along ⟶ AB .
5. Draw and label the image of PQ under a translation along ⟶ EF .
6. Consider the diagram of intersecting lines, ⟷ AB and ⟷ BC , and a vector, ⟶ GH .
a. Draw and label the images of ⟷ AB and ⟷ BC under a translation along ⟶ GH
b. Describe some relationships you observe between the figure and its image in the diagram. You may want to comment on angles or lines.
7. Consider the diagram of ⟶ UP and figure MATH, which includes rectangle MATH and a semicircle with diameter MA .
a. Draw and label the image of figure MATH under a translation along ⟶ UP
b. Describe some relationships you observe between the figure and its image in the diagram. You may want to comment on the corresponding parts, which include the corresponding sides and the corresponding angles.
Properties of Translations
For problems 8–12, determine whether the statement is always, sometimes, or never true. Provide reasoning and an example or nonexample from problems 4–7 to support your claim.
8. A translation changes the length of a segment.
9. A translation maps a line to a parallel line.
10. A translation maps an angle to an angle of equal measure.
11. A translation maps a line to a line.
12. A translation maps parallel lines to parallel lines.
A Rigid Motion—Translation
A translation along maps a figure to its image.
vector name
Example: A translation along maps P to P′ .
Student Edition: Grade 8, Module 2, Topic A, Lesson 2
Name Date
1. Draw and label the image of figure ABCD under a translation along ⟶ KL
2. Under a translation, △ A′ B ′C ′ is the image of △ ABC. Label
with all known segment lengths and angle measures.
Student Edition: Grade 8, Module 2, Topic A, Lesson 2
Name Date
Translations
In this lesson, we
• translated along a vector to map a figure to its image.
• identified that under a translation
▸ lines map to lines,
▸ segments map to segments of the same length,
▸ angles map to angles of the same measure, and
▸ parallel lines map to parallel lines.
Examples
1. Describe the translation with precise language.
The arrowhead for ⟶ TW shows a movement down and to the left.
To map figure A onto figure A′, the vector needs to go up and to the right.
Terminology
A vector is a directed line segment. The direction of ⟶ AB is determined by starting at point A, moving along the segment, and ending at point B. This direction is shown by an arrowhead placed at point B.
A translation is a rigid motion along a vector that maps a figure to its image. A translation along ⟶ AB maps point P to a point P ′ with the following features:
• The distance from P to P ′ is equal to the length of ⟶ AB
• The direction of ⟶ PP′ is the same as the direction of ⟶ AB
• If P is not on ⟷ AB , then the path from P to P ′ is parallel to ⟶ AB
• If P is on ⟷ AB , then P ′ is also on ⟷ AB
The arrowhead for ⟶ X M shows a movement up and to the right.
A translation along ⟶ X M maps figure A onto figure A′ .
2. Draw and label the image of figure ABCD under a translation along the given vector. Label any known segment lengths and angle measures.
• Extend ⟶ RS
• Trace ⟶ RS and figure ABCD onto a transparency.
• Slide the transparency along ⟷ RS until point R is on point S
• Lift the transparency to mark the locations of the image vertices on the page and label the vertices.
• Use a straightedge to connect the image vertices.
Student Edition: Grade 8, Module 2, Topic A, Lesson 2
Name
For problems 1–4, name the vector that maps the figure onto the image provided. Then describe the translation.
For problems 5–10, draw and label the image of each figure under a translation along ⟶ RW .
For problems 11 and 12, draw and label the image of each figure under a translation along the given vector. Label any known segment lengths and angle measures.
12.
13. Under a translation along a vector, will a figure and its image ever intersect?
14. Under a translation along a vector, will the image A′B′ always be parallel to AB ? Remember
For problems 15–18, evaluate.
19. What is the side length of a square with an area of 81 square units?
20. Plot the points in the coordinate plane.
(3, 5), (5, 2), (6, 4), (7, 2), (9, 5)
Student Edition: Grade 8, Module 2, Topic A, Lesson 3
Name Date
Reflections
Reflect
1. Draw and label the image of figure ABCD under a reflection across line ��.
2. Draw and label the image of figure ABCDE under a reflection across ⟷ GH .
3. Draw and label the image of figure AB under a reflection across line ��.
4. Draw and label the image of △ ACE under a reflection across ⟷ RS .
5. Draw and label the image of figure FILM under a reflection across line ��.
Another Rigid Motion—Reflection
A reflection across maps a figure to its image. line name
Example: A reflection across maps P to P′ .
Student Edition: Grade 8, Module 2, Topic A, Lesson 3
Name Date
Draw and label the image of △ ABC under a reflection across line ��
Student Edition: Grade 8, Module 2, Topic A, Lesson 3
Name Date RECAP
Reflections
In this lesson, we
• reflected across a line to map a figure to its image.
• identified that under a reflection
▸ lines map to lines,
▸ segments map to segments of the same length,
▸ angles map to angles of the same measure, and
▸ parallel lines map to parallel lines.
Examples
1. Describe the reflection shown in the diagram with precise language.
Terminology
A reflection is a rigid motion across line ��, called the line of reflection, that maps a figure to its image. A reflection across line �� maps point P to a point P′ with the following features:
• P and P′ are on opposite sides of ��.
• The distance from P to line �� is equal to the distance from P′ to ��.
• A line passing through P and P′ is perpendicular to ��.
Because points A and B are on the line of reflection, they are in the same location under the reflection. So they have two labels.
A reflection across line �� maps figure FLIPS onto figure F′L′I′P′S′, point A to point A′, and point B to point B′
• If P is on the line of reflection, then P and P′ are the same point. P ʹ P 𝓁
2. Draw and label the image of △ PQR under a reflection across line ��. Label any known segment lengths and angle measures.
• Draw and label a point O on line ��
• Trace line ��, the point O on line ��, and △ PQR on a transparency.
• Flip the transparency.
• Line up point O and line ��.
• Lift the transparency to mark the locations of the image vertices on the page, and label the vertices.
• Use a straightedge to connect the image vertices.
Student
Edition: Grade 8, Module 2, Topic A, Lesson 3
Name
For problems 1–6, draw and label the image of the point or figure under a reflection across line ��
For problems 7–9, describe the reflection that maps the figure onto the image provided.
10. A reflection across line �� is shown in the diagram.
a. Fill in the boxes with the missing labels.
b. Label any known segment lengths and angle measures in the diagram.
c. What is the measure of ∠IJK ? ∠KIJ ? ∠ ABC ? How do you know?
d. What is the length of the image of FH and the length of IK ? How do you know?
e. What is the location of the image of point D under a reflection across line �� ? Explain.
11. Describe the rigid motion that maps circle C onto circle C ′. How do you know?
Remember
For problems 12–15, evaluate.
16. A right triangle has leg lengths of 8 units and 10 units. What is the length of the hypotenuse?
17. In the given diagram, two lines meet at a point that is also the endpoint of a ray.
a. What angle relationship would help you solve for x ?
b. Determine the measure of ∠ EFD.
Student Edition: Grade 8, Module 2, Topic A, Lesson 4
Name Date
Translations and Reflections on the Coordinate Plane
Translations on the Coordinate Plane
1. Circle all the correct ways to describe the translation that maps △ ABC onto △ A′B′C′
A translation 2 units down and 4 units left A translation along ⟶ AB′
A translation 4 units down and 2 units left A translation 4 units left and 2 units down
A translation along ⟶ AA′ A translation 2 units left and 4 units down
2. Graph and label the image of figure ABCD under a translation 6 units right and 2 units down.
Reflections on the Coordinate Plane
3. Dylan made an error graphing and labeling the image of figure ABCD under a reflection across the x-axis.
a. Describe Dylan’s error.
b. Graph and label the correct image of figure ABCD under a reflection across the x-axis. Then label the coordinates for each vertex.
4. Graph and label the image of figure ABCD under a reflection across the y-axis. Then label the coordinates for each vertex of the image.
B(2, 4) C(4, 4)
Student Edition: Grade 8, Module 2, Topic A, Lesson 4
Name Date
1. Graph and label the image of figure ABCDE under a translation 7 units left and 6 units up.
2. Graph and label the image of △ JKL under a reflection across the y-axis.
Student Edition: Grade 8, Module 2, Topic A, Lesson 4
Name Date
Translations and Reflections on the Coordinate Plane
In this lesson, we
• applied translations and reflections on the coordinate plane.
• used coordinates to describe the location of an image under a translation or a reflection.
Examples
1. Graph and label the image of figure JKLM under a translation 6 units down and 3 units right.
Each point of the figure maps to a point of its image that is 6 units down and 3 units right.
2. Graph and label the image of △ ABC under a reflection across the x-axis.
Point A is on the line of reflection, so it maps to point A′ in the same location.
Each point of the triangle maps to a point that is the same distance from the line of reflection, the x-axis.
Student Edition: Grade 8, Module 2, Topic A, Lesson 4
Name Date
For problems 1–4, graph and label the image of the figure under the given translation.
1. 6 units right
3. 2 units left and 5 units up
5. Consider quadrilaterals ABCD, EFGH, and IJKL.
a. Which figure is the image of quadrilateral ABCD under a translation? Describe the translation.
b. Which figure is the image of quadrilateral ABCD under a reflection? Describe the reflection.
For problems 6–9, graph and label the image of the figure under a reflection across the given line.
6. x-axis
7. y-axis
8. y-axis
9. x-axis
10. The point A′(7, 9) is the image of point A(2, 2) under a translation. Which of the following describes the translation?
A. 5 units down and 11 units right
B. 5 units left and 11 units up
C. 5 units right and 11 units down
D. 5 units up and 11 units left
11. Determine whether the following statement is always, sometimes, or never true. Explain your reasoning.
Under a reflection across the x-axis, the image of point (x, y) has the coordinates (x, y).
Remember
For problems 12–15, evaluate.
12. 2 3 + (−2)
2 3 − (−2)
14. 2 3 (−2)
2 3 ÷ (−2)
16. If the length of CD is 5 units, what is the length of C′D′ under a translation?
17. If the measure of ∠ EBA is 75°, what is the measure of ∠ E′B′A′ under a rotation?
18. The vertices of a triangle are located at ( 4, 3), ( 4, 7), and (3, 7). Plot and label the vertices. Then draw the triangle.
Student Edition: Grade 8, Module 2, Topic A, Lesson 5
Name Date
Rotations
1. Draw and label the image of OP under a rotation around point O.
Rotate
For problems 2 and 3, draw and label the image of the figure under the given rotation around point O.
2. 90° clockwise
3. 180° counterclockwise
A Third Rigid Motion—Rotation
A rotation around maps a figure to its image.
number of degrees direction center of rotation
Example: A rotation around maps P to P′ .
4. Consider parallelogram DEFG and point O.
a. Draw and label the image of parallelogram DEFG under a 90° counterclockwise rotation around point O.
b. What is the length of D′G′ ? Explain your reasoning.
c. What is the measure of ∠F′? Explain your reasoning.
5. Consider △ AOB
a. Draw and label the image of △ AOB under a 45° clockwise rotation around point O.
b. What is the length of A′B′ ? Explain your reasoning.
c. What is the measure of ∠O′A′B′? Explain your reasoning.
6. Determine whether each statement is true or false.
a. In problem 2, MN is parallel to M′N′ .
b. In problem 2, MN is the same length as M′N′
c. In problem 3, the measure of ∠D′E′F′ is greater than the measure of ∠DEF.
d. In problem 4, DE is the same length as E′F′
e. In problem 4, D′E′ is parallel to F′G′ .
f. In problem 4, OF is the same length as OF′
g. In problem 5, the measure of ∠ A′O′B′ is equal to the measure of ∠ AOB.
h. In problem 5, OB is the same length as OB′
Reflection or Rotation?
7. Analyze the rectangles in the table to identify the rigid motion that maps rectangle ABCD onto its image.
Student Edition: Grade 8, Module 2, Topic A, Lesson 5
Name
Consider figure ABC and point O
a. Draw and label the image of figure ABC under a 90° clockwise rotation around point O
b. Include any known segment lengths and angle measures.
c. How do you know your measurements in the image of figure A′B′C′ are correct?
Student Edition: Grade 8, Module 2, Topic A, Lesson 5
Name Date
Rotations
In this lesson, we
• rotated clockwise and counterclockwise around a point to map a figure to its image.
• identified that under a rotation
▸ lines map to lines,
▸ segments map to segments of the same length,
▸ angles map to angles of the same measure, and
▸ parallel lines map to parallel lines.
Example
Draw and label the image of figure STUV under a 90° counterclockwise rotation around point O Label any known segment lengths and angle measures.
• Trace point O and figure STUV on a transparency.
• Keeping the traced point O lined up with point O on the page, rotate the transparency 90°, or one-quarter turn, counterclockwise.
• Lift the transparency to mark the locations of the image vertices on the page and label the vertices.
• Use a straightedge to connect the image vertices.
Terminology
A rotation is a rigid motion counterclockwise or clockwise by a given number of degrees around a point, called the center of rotation, that maps a figure to its image. A d° counterclockwise (or clockwise) rotation around point O maps any point P that is not O to a point P′ with the following features:
• P ′ is located counterclockwise (or clockwise) from P on a circle centered at O with radius OP .
• The measure of ∠POP ′ is d°.
• The center of rotation O and its image O ′ are the same point. P
Clockwise
Counterclockwise
Student Edition: Grade 8, Module 2, Topic A, Lesson 5
Name
For problems 1 and 2, label the image of the figure under the given rotation around point O 1. 180° clockwise
45° counterclockwise
For problems 3–8, draw and label the image of the point or figure under the given rotation around point O
3. 90° clockwise
180° clockwise
5. 90° counterclockwise
6. 270° counterclockwise
7. 45° counterclockwise
8. 180° clockwise
9. Consider AB , ∠CDE, point F, and point O.
F
a. Draw and label the images of the figures and point under a 90° clockwise rotation around point O.
b. What is the length of A′B′
c. What is the measure of ∠C ′D ′E ′ ?
10. Ethan says a rotation can map a figure onto itself. Do you agree with Ethan? Explain.
Remember
For problems 11–14, evaluate.
11. 6 + 2 3
6 − 2 3 13. 6 · 2 3
15. Solve the equation x 2 = 17. Identify all solutions as rational or irrational.
16. Plot the points in the coordinate plane. (0, 4), (−4, 0), (−3, 1), (0, 0), (−1, −3), (3, 2), (2, −3)
Student Edition: Grade 8, Module 2, Topic A, Lesson 6
Name
Rotations on the Coordinate Plane
1. Which rigid motion is shown? Explain.
Rotating Around the Origin
For problems 2–5, graph and label the image of the point or figure under the given rotation around the origin. Then identify the coordinates of the point, endpoints, or vertices of the image.
2. 90° counterclockwise
3. 90° clockwise
270° clockwise
5. 180° counterclockwise
6. Each of the rotations in problems 2–5 can be described in the other direction. For each problem, write another way to describe the rotation around the origin.
Problem 2:
Problem 3:
Problem 4:
Problem 5:
Rotating 180° Around the Origin
7. Use the given coordinate plane and table with the following problems.
a. Graph a figure with 4 vertices. Label the vertices.
b. Graph and label the image of the figure under a 180° rotation around the origin.
c. Complete the table for the figure and its image.
Figure Name
Vertex Coordinates of the Figure
Vertex Coordinates of Its Image
d. Compare the corresponding vertex coordinates of the figure and its image. Make a conjecture about the relationship between the coordinates of a point and the coordinates of its image under a 180° rotation around the origin.
8. What are the coordinates of the image of a point (x, y) under a 180° rotation around the origin?
Parallel or Not Parallel
9. Determine whether the statement is always, sometimes, or never true. A 180° rotation around the origin maps line �� to a line parallel to line ��.
Test Case 1: Line �� is parallel to the x-axis.
Test Case 2: Line �� is parallel to the y-axis.
Test Case 3: Line �� passes through the origin.
Student Edition: Grade 8, Module 2, Topic A, Lesson 6
Student Edition: Grade 8, Module 2, Topic A, Lesson 6
Name Date
1. Plot and label the image of point A under a 90° counterclockwise rotation around the origin. Then identify the coordinates of the image.
2. Plot and label the image of point B under a 180° rotation around the origin. Then identify the coordinates of the image.
Student Edition: Grade 8, Module 2, Topic A, Lesson 6
Name
Rotations on the Coordinate Plane
In this lesson, we
• applied rotations around the origin on the coordinate plane.
• used coordinates to describe the location of an image under a rotation.
Examples
Graph and label the image of the point or figure under the given rotation around the origin.
1. 90° clockwise
Trace point J, the origin, and the axes on the transparency.
After every quarter turn, align the traced axes with the axes on the page. This helps keep the angle of rotation and the location of the image exact.
2. 270° counterclockwise
A 270° counterclockwise rotation maps a point to the same location as a 90° clockwise rotation.
3. 180° counterclockwise
The coordinates of the image of a point under a 180° rotation around the origin are opposites of the coordinates of the original point.
For example, point F has coordinates (9, −7) and point F ′ has coordinates (−9, 7)
A 180° counterclockwise rotation maps a point to the same location as a 180° clockwise rotation.
Student Edition: Grade 8, Module 2, Topic A, Lesson 6
Name Date
For problems 1–6, graph and label the image of the point or figure under the given rotation around the origin.
1. 90° counterclockwise
3. 180° clockwise
4.
counterclockwise
5. 90° clockwise
6. 180° counterclockwise
7. Yu Yan misreads problem 6 and rotates 180° clockwise instead of 180° counterclockwise. What are the coordinates of the vertices for Yu Yan’s image? Explain.
8. Which rotations around the origin map ∠ BAC onto ∠ B′A′C′? Choose all that apply.
9. Which rotations around the origin map quadrilateral ABCD onto quadrilateral A′B′C ′D′? Choose all that apply.
A. 90° clockwise
B. 90° counterclockwise
C. 180° clockwise
D. 180° counterclockwise
E. 270° clockwise
F. 270° counterclockwise
A. 90° clockwise
B. 90° counterclockwise
C. 180° clockwise
D. 180° counterclockwise
E. 270° clockwise
F. 270° counterclockwise
For problems 10–13, determine the coordinates of the image of the given point under a 180° rotation around the origin.
P (5, 0)
D (−6, 6)
Remember
For problems 14–17, evaluate.
M (8, 10)
B (−4, 7)
18. Draw the image of ∠ ABC under a translation along ⟶ DE . Label your image with the correct endpoints, segment lengths, and angle measures.
10.
11.
12.
13.
19. Consider the given diagram where two lines meet at a point.
a. What angle relationship would help you solve for x ?
b. Find the value of x
Rigid Motions and Congruent Figures
Student Edition: Grade 8, Module 2, Topic B
Congruent Twins
TOPIC B
Two buttons from the same factory.
Two T-shirts of the same size and style.
Two unopened tubes of the same toothpaste.
In ordinary speech, we might call these identical. But mathematicians prefer a different word. It’s a word for any pair of shapes that look the same but are distinct.
That word is congruent
How do we know if two things are congruent? One way is to see whether you could place one shape exactly on top of the other by using only a sequence of translations, reflections, and rotations. If so, that means all the parts must line up: corner to corner, side to side, and angle to angle.
For problems 1–4, complete the table by describing the rigid motion that maps the original figure onto its image and the rigid motion that maps the image back onto its original figure.
• drew and labeled images of figures under rigid motions.
• described which rigid motion maps an image back onto its original figure.
Examples
1. Figure PQRS is shown.
a. Graph and label the image of figure PQRS under a reflection across the y-axis.
b. Describe the rigid motion that maps figure P ′Q ′R ′S ′ back onto figure PQRS. A reflection across the y-axis maps figure P ′Q ′R ′S ′ back onto figure PQRS. A reflection across the same line maps an image back onto its original figure.
2. In the diagram, figure S and ⟶ EF are shown.
S ʹ S E F
a. Draw and label the image of figure S under a translation along ⟶ EF .
b. Describe the rigid motion that maps figure S ′ back onto figure S. A translation along ⟶ FE maps figure S ′ back onto figure S.
3. Point D, △ ABC, and △ A′B ′C ′ are shown.
A translation with the same distance but in the opposite direction maps an image back onto its original figure.
a. Describe a rigid motion that maps △ ABC onto △ A′B ′C ′
A 90° clockwise rotation around point D maps △ ABC onto △ A′B ′C ′
A 270° counterclockwise rotation around point D maps △ ABC onto △ A′B ′C ′
b. Describe a rigid motion that maps △ A′B ′C ′ back onto △ ABC
Both rigid motions are correct because the sum of the measures of the angles of rotation, 90° and 270°, is 360°.
A 90° counterclockwise rotation around point D maps △ A′B ′C ′ back onto △ ABC. A 270° clockwise rotation around point D maps △ A′B ′C ′ back onto △ ABC.
a. Draw and label the images of these points under a reflection across line ��.
b. Describe the rigid motion that maps points A′ , B ′, and C ′ back to points A, B, and C.
2. Figure E and ⟶ FG are shown.
a. Draw and label the image of figure E under a translation along ⟶ FG .
b. Describe the rigid motion that maps figure E ′ back onto figure E.
3. Parallelogram EFGH is shown.
a. Graph and label the image of parallelogram EFGH under a 180° rotation around the origin.
b. Describe the rigid motion that maps parallelogram E ′F ′G ′H ′ back onto parallelogram EFGH.
4. Curve ST is shown.
a. Graph and label the image of curve ST under a translation 1 unit down and 4 units right.
b. Describe the rigid motion that maps curve S ′T ′ back onto curve ST
5. Point P, rectangle KLMN, and rectangle K ′L ′M ′N ′ are shown.
a. Describe the rigid motion that maps rectangle KLMN onto rectangle K ′L ′M ′N ′ .
b. Describe the rigid motion that maps rectangle K ′L ′M ′N ′ back onto rectangle KLMN
6. Line ��, △LMN, and △L ′M ′N ′ are shown.
a. Describe the rigid motion that maps △LMN onto △L ′M ′N ′ .
b. Describe the rigid motion that maps △L ′M ′N ′ back onto △LMN
7. In the diagram, ⟶ OX , △LMN, and △L ′M ′N ′ are shown.
a. Describe the rigid motion that maps △LMN onto △L ′M ′N ′ .
b. Describe the rigid motion that maps △L ′M ′N ′ back onto △LMN
8. A 45° clockwise rotation around point P maps △XYZ onto △X ′Y ′Z ′. Describe a rigid motion that maps △X ′Y ′Z ′ back onto △XYZ.
9. A translation 3 units left and 2 units up on the coordinate plane maps point F to point F ′ . Describe the rigid motion that maps point F ′ back to point F.
Remember
For problems 10–13, multiply.
10. 5 a(1 5 )
11. 6 x(1 6 )
12. 11 b( 1 11 )
14. In the diagram, △ABC and line �� are shown.
13. 15 z ( 1 15 )
a. Draw and label the image of △ABC under a reflection across line ��.
b. What is the measure of ∠A′B ′C ′?
c. What is the length of A ′ C ′ ?
15. Consider the given diagram where two lines intersect at point B.
(x + 8)° (15x – 52)°
a. What angle relationship would help you solve for x ?
b. Find the measure of ∠ABD and the measure of ∠ABC
1. Draw and label the image of CD under the following sequence of rigid motions.
• 90° clockwise rotation around point C
• Translation along ⟶ AB
2. Draw and label the image of △XYZ under the following sequence of rigid motions.
• Translation along ⟶ RS
• Reflection across line ��
3. Graph and label the image of figure QRST under the following sequence of rigid motions.
• 270° clockwise rotation around the origin
• Translation 3 units up and 2 units left
4. Graph and label the image of figure S under the following sequence of rigid motions.
• Translation 2 units up and 6 units left
• 90° clockwise rotation around the origin
• Reflection across the y-axis
a. Graph and label the image of figure EFGH under the following sequence of rigid motions.
• Reflection across the x-axis
• Reflection across the y-axis
b. Can a single rigid motion map figure EFGH onto its image? If so, describe the single rigid motion.
5. Figure EFGH is shown.
For problems 6–8, describe a sequence of rigid motions that maps the figure onto its image. Your sequence may have more than one rigid motion even though the vertices of the image are labeled with a single prime.
6. Describe a sequence of rigid motions that maps △DEF onto △
7. Describe a sequence of rigid motions that maps figure HOME onto figure H ′O′M ′E ′ .
8. Describe a sequence of rigid motions that maps △LOG onto △L′O ′G ′ .
9. Use figure A and point P to answer the following questions.
a. Describe a sequence of rotations that maps figure A back onto itself.
b. Describe a single rotation that maps figure A back onto itself.
Remember
For problems 10–13, multiply.
10. 5 a (2 5 ) 11. 10 x ( 7 10)
12. 7 c (3 7 ) 13. 4 g (− 5 4 )
14. Figure A′B ′C ′ is the image of figure ABC under a rotation around point O.
a. What is the measure of A ′ C ′ ?
b. What is the measure of ∠B ′A′C ′?
15. The measure of ∠ ABC is 30°.
a. If ∠ ABC and ∠CBE are complementary, what is the measure of ∠CBE ?
b. If ∠ ABC and ∠CBF are supplementary, what is the measure of ∠CBF ?
1. Will figure ABCD map onto figure EFGH by using the following sequence of rigid motions?
• Apply a 90° counterclockwise rotation around point X. Point X is located directly below point C.
• Apply a reflection across line ��. Line �� is a vertical line to the right of CD .
Figures That Touch
For problems 2 and 3, describe a sequence of rigid motions that shows the given mapping. Draw any vectors, lines of reflection, or centers of rotation that are needed.
2. Map △ ABC onto △ DBC
3. Map △ ABC onto △ DBE
4. Consider △ ABC and △ DBE. Describe a sequence of rigid motions that shows the given mapping. Draw any vectors, lines of reflection, or centers of rotation that are needed.
B C E
a. Map △ ABC onto △ DBE.
b. Map △ DBE onto △ ABC.
Figures That Are Separate
For problems 5–8, describe the sequence of rigid motions that maps one figure onto a congruent figure. Draw any vectors, lines of reflection, or centers of rotation that are needed.
For problems 1 and 2, identify which of the given sequences of rigid motions map one figure in the diagram onto the congruent figure. Choose all that apply.
A. A 180° rotation around point A
B. A translation along ⟶ CD followed by a reflection across the line containing DE
C. A 90° clockwise rotation around point A followed by a reflection across the line containing AE
D. A reflection across the line containing AC followed by a reflection across the line containing AD
1. △ ABC ≅ △ ADE
2.
A. A translation along ⟶ ER , a 90° clockwise rotation around point R, and a reflection across ⟷ RK
B. A translation along ⟶ ER , a 90° counterclockwise rotation around point R, and a reflection across ⟷ RK
C. A translation along ⟶ PM , a 90° clockwise rotation around point M, and a reflection across ⟷ MR
D. A translation along ⟶ PM , a 90° counterclockwise rotation around point M, and a reflection across ⟷ MR
For problems 3–6, describe a sequence of rigid motions that maps one figure onto the congruent figure.
△ PEN ≅ △ MRK
3. Figure BLING ≅ figure SPARG
4. △ CAT ≅ △ COT
5. Square ABCD ≅ square EFGH
6. Rectangle MATH ≅ rectangle SUBR
7. Describe the sequence of rigid motions that maps △ ABC onto △ XYZ.
8. The figures in the diagram are congruent. Maya and Ethan each write a sequence of rigid motions that maps one figure onto the other.
Maya: A translation along ⟶ DC followed by a reflection across ⟷ CA maps figure DOG onto figure CAT.
Ethan: A translation along ⟶ AG followed by a 90° clockwise rotation around point G maps figure CTA onto figure DOG.
Who is correct? Why?
9. Describe the sequence of rigid motions that maps figure MAIL onto figure SPOT.
Remember
For problems 10–13, write an equivalent expression.
10. 3(x 2) 11. −5(x 4)
12. −7(x + 3) 13. 8( x + 7)
14. A rigid motion maps figure ABCD onto figure A′B′C′D′. What rigid motion maps figure A′B′C′D′ back onto figure ABCD?
1. Do you think the tabletops are congruent? Use a transparency to check, and then explain your reasoning.
All Things Being Equal
For problems 2 and 3, use a transparency to determine whether the given figures are congruent. If so, describe a sequence of rigid motions that maps one figure onto the other. If not, explain how you know.
2. Figure GOLD and figure FISH
3. △ COW and △ HEN
Design Thinking
4. Circle two congruent figures in the rug. Write a sequence of rigid motions to show that the two figures are congruent. Draw any needed vectors, lines of reflection, or points of rotation.
Use a transparency to determine whether figure GHJK and figure WTUV are congruent. If so, describe a sequence of rigid motions that maps one figure onto the other. If not, explain how you know.
• identified two figures as congruent or not congruent.
• showed that two figures are congruent by describing a sequence that maps one figure onto the other.
• explained that two figures are not congruent when the distances between points in the figure are not the same as the distances between corresponding points in the image.
Examples
Use a transparency to determine whether the given figures are congruent. If so, describe a sequence of rigid motions that maps one figure onto the other. If not, explain how you know.
1. △ HAT and △ SUN
Find a sequence of rigid motions that maps △HAT onto △SUN to show the triangles are congruent.
△ HAT and △ SUN are congruent. A translation along ⟶ AU maps point A to point U. Then a 180° rotation around point U maps △ HAT onto △ SUN
Use a transparency to see that FY ≠ WX and FL ≠ WA.
Count grid spaces to show that FY ≠ WX. Use the Pythagorean theorem to show that FL ≠ WA.
△FLY and △WAX are not congruent because the distances between points in △FLY are not equal to the distances between corresponding points in △WAX. I can use my transparency to see that FY ≠ WX and FL ≠ WA.
1. Which sequence of rigid motions shows △ ABC ≅ △ RST ? Circle your answer.
First Rigid Motion
Reflection across the x-axis
90° clockwise rotation around the origin
Translation 2 units up
Translation 10 units up and 2 units left
Second Rigid Motion
Translation 2 units up
Reflection across the x-axis
Translation 10 units up and 2 units left
90° clockwise rotation around the origin
2. In the diagram, figure E is congruent to figure D. Which sequences of rigid motions describe how to map one figure onto the other? Choose all that apply.
A. A translation along ⟶ AB followed by a reflection across line �� maps figure E onto figure D.
B. A translation along ⟶ BA followed by a reflection across line �� maps figure D onto figure E.
C. A reflection across line �� followed by a translation along ⟶ AB maps figure E onto figure D.
D. A reflection across line �� followed by a translation along ⟶ BA maps figure E onto figure D.
E. A reflection across line �� followed by a translation along ⟶ BA maps figure D onto figure E.
For problems 3–5, use a transparency to determine whether the given figures are congruent. If so, describe a sequence of rigid motions that maps one figure onto the other. If not, explain how you know.
3. Figure ABC and figure DEF
4. Figure STUV and figure DEFG
5. △ RST and △ NLE
6. Figures J and K are shown.
a. Is figure J congruent to figure K ? Explain.
b. Is figure K congruent to figure J ? Explain.
7. A sequence of rigid motions creates the following pattern.
Quadrant II
Quadrant III y x
Quadrant I
Quadrant IV
Match each statement with the sequence of rigid motions that shows the patterns are congruent. A sequence may be used more than once.
Statement
The pattern in Quadrant I is congruent to the pattern in Quadrant II.
The pattern in Quadrant II is congruent to the pattern in Quadrant III.
The pattern in Quadrant III is congruent to the pattern in Quadrant IV.
The pattern in Quadrant IV is congruent to the pattern in Quadrant I.
The pattern in Quadrant I is congruent to the pattern in Quadrant III.
Sequence of Rigid Motions
A 180° rotation around the origin
A 90° counterclockwise rotation around the origin
A reflection across the y-axis followed by a 90° counterclockwise rotation around the origin
A 90° counterclockwise rotation around the origin followed by a reflection across the y-axis
A reflection across the x-axis
Remember
For problems 8–11, write an equivalent expression.
12. Plot and label the image of point A under a 90° rotation counterclockwise around the origin. Then identify the coordinates of the image.
8. 3(x + 2)
9. 7(x + 1)
10. 5(x − 5)
11. 8(x − 3)
13. Consider a segment that has endpoints (5, −3) and (−4, −3).
a. Plot the points and create the segment in the coordinate plane.
b. What is the length of the segment?
Angle Relationships
Student Edition: Grade 8, Module 2, Topic C
Battle of the Triangles
TOPIC C
When you look at the world of triangles, you see a lot of very different faces. Some are skinny. Some are wide. Some are symmetric. Some are not. They differ in so many ways, so one might think they should also differ in the sum of their angle measures. Right?
Geometry is a powerful force. It can take trillions of different shapes and unify them all under the same rule. Every single triangle—skinny, wide, symmetric, whatever—has three angles, and the measures of those three angles will always add up to exactly the same total, no matter what.
For problems 1 and 2, record which angle pairs you think are congruent.
1. Line ℯ intersects lines �� and ��.
2. Line �� intersects lines �� and ��.
Angle Relationships
3. Complete the angle relationships table as a class.
Description:
Line �� is parallel to line ��.
Description:
Line �� is parallel to line ��
Description: Description:
Line �� is parallel to line ��.
Line �� is parallel to line ��.
Finding Unknown Angle Measures
4. In the diagram, line �� intersects parallel lines �� and ��.
a. If possible, find the measures of the remaining seven angles. Write those measures on the diagram.
b. Complete the table. Write the measure of each angle from part (a). Then for each angle measure, provide an explanation of that angle’s relationship to ∠ ABC.
Angle Name Angle Measure
∠DBC
∠DBF
∠FBA
∠EFB
∠GFB
∠GFH
∠EFH
Explanation
c. Use rigid motions to explain why each angle is congruent to ∠ABC.
Angle Name Rigid Motion
DBF
EFB
GFH
5. In the diagram, ⟷ FT intersects parallel lines, ⟷ EK and ⟷ CR .
a. If possible, find the measures of the remaining seven angles. Write those measures on the diagram.
b. Name all the pairs of corresponding angles.
c. Name all the pairs of alternate interior angles.
d. Name all the pairs of alternate exterior angles.
6. In the diagram, the lines ℊ, ��, and �� are shown.
a. If possible, find the measures of the remaining seven angles. Write those measures on the diagram.
b. Choose one angle that is congruent to ∠VET and describe the rigid motion that maps that angle onto ∠VET.
c. What is the angle relationship between ∠VET and ∠ERY?
d. Can you determine the measure of ∠ERY? Explain.
• used rigid motions to show that an angle pair was congruent.
• found the measure of an angle based on its relationship with another angle.
Terminology
Given a pair of lines �� and �� in a plane, a third line �� is a transversal if it intersects line �� at a single point and intersects line �� at a single but different point.
If the lines cut by the transversal are parallel, then these angles are congruent.
Given lines �� and �� and transversal �� shown in the diagram, there are
• four pairs of corresponding angles: ∠1 and ∠5,
and
and
• two pairs of alternate interior angles:
and
5; and • two pairs of alternate exterior angles:
Example
In the diagram, line ℊ intersects parallel lines �� and ��.
a. What is the angle relationship between ∠1 and ∠5?
They are corresponding angles.
b. Describe a sequence of rigid motions that maps ∠1 onto ∠5.
A translation along ⟶ DF maps ∠1 onto ∠5.
c. What is the angle relationship between ∠3 and ∠5?
They are alternate interior angles.
d. Describe a sequence of rigid motions that maps ∠3 onto ∠5.
A translation along ⟶ DF followed by a 180° rotation around point F maps ∠3 onto ∠5.
e. If the measure of ∠3 is 56°, what is the measure of ∠7? Explain. 56°
Because lines �� and �� are parallel, ∠1 can be mapped onto ∠5. That means the corresponding angles are congruent.
Because lines �� and �� are parallel, corresponding angles are congruent, so they are equal in measure.
f. If the measure of ∠4 is 124°, what is the measure of ∠6? Explain. 124°
Because lines �� and �� are parallel, alternate exterior angles are congruent, so they are equal in measure.
• determined that the sum of the interior angle measures of a triangle is 180°.
• found the measure of one interior angle when given the other two interior angle measures of a triangle.
Examples
1. Find the measure of ∠ ACB
2. Find the measure of ∠ EFD.
32° + 48° = 80° 180° − 80° = 100° m ∠ ACB = 100°
Add the two given angle measures. Then subtract that sum from 180°
90° + 63° = 153°
63°
180° − 153° = 27° m ∠ EFD = 27° m ∠ DEF = 90°
Terminology
An interior angle of a polygon is an angle formed by two adjacent sides of the polygon. For example, ∠1, ∠2, ∠3, ∠4, and ∠5 are all interior angles of the pentagon shown in the diagram. 1
Given lines �� and �� and transversal �� shown in the diagram, there are two pairs of same-side interior angles: ∠3 and ∠5, ∠4 and ∠6.
3. Find the measure of ∠GHJ
The two remaining angle measures are equal. Divide the difference by 2 to find the measure of each angle.
180° 22° = 158° 158° 2 = 79° m ∠ GHJ = 79°
Subtract 22° from 180° to find the sum of the other two angle measures.
• found that if corresponding, alternate interior, or alternate exterior angles created by two lines cut by a transversal are congruent, then the two lines are parallel.
• used pairs of angle measures to determine whether two lines are parallel.
Examples
For problems 1–3, use the diagram of lines ��, ��, and ��.
a. Determine whether the given angle measures show that lines �� and �� are parallel or not parallel. If not enough information is given, write undetermined.
b. Explain your answer for part (a).
Corresponding Angles
• ∠1 and ∠3
• ∠6 and ∠8
• ∠2 and ∠4 • ∠5 and ∠7
Alternate Interior Angles
• ∠2 and ∠6 • ∠3 and ∠7
Alternate Exterior Angles
• ∠1 and ∠5 • ∠4 and ∠8
1. m∠5 = 110° , m∠7 = 110°
a. Parallel
b. Because ∠5 and ∠7 are corresponding angles and have the same measure, lines �� and �� are parallel.
2. m∠3 = 118° , m∠6 = 62°
a. Undetermined
b. Because the angles are on a line and formed by the intersection of lines �� and ��, there is not enough information to determine the relationship between lines �� and ��.
3. m∠2 = 43° , m∠6 = 44°
a. Not parallel
b. Because ∠2 and ∠6 are alternate interior angles that do not have the same measure, lines �� and �� are not parallel.
4. The diagram of ⟷ AB , ⟷ GF , ⟷ CH , and ↔ EI is shown.
∠ECD and ∠BEF are congruent.
a. Are ⟷ CH and ↔ EI parallel? Explain.
Yes. ⟷ CH and ↔ EI are parallel because corresponding angles, ∠ECD and ∠BEF, are congruent.
b. Are ⟷ AB and ⟷ GF parallel? Explain.
m∠CDF + m∠ FDH = 180°
50° + m∠ FDH = 180°
m∠ FDH = 130°
∠CDF and ∠ FDH are a linear pair, so the sum of their measures is 180°
Yes. ⟷ AB and ⟷ GF are parallel because the corresponding angles, ∠ECD and ∠FDH, are congruent.
For problems 1–7, use the diagram of lines ℊ, ��, and ��. Determine whether the given angle measures show that lines ℊ and �� are parallel or not parallel. If there is not enough information, choose Undetermined.
1. m∠4 = 56°, m∠2 = 56°
2. m∠3 = 124°, m∠6 = 124°
3. m∠8 = 124°, m∠4 = 56°
4. m∠5 = 56°, m∠2 = 56°
5. m∠6 = 124°, m∠7 = 56°
6. m∠5 = 56°, m∠7 = 55°
7. m∠1 = 124°, m∠4 = 56°
8. Jonas says there is not enough information to determine whether ⟷ AB is parallel to ⟷ CD . Do you agree with Jonas? Explain.
9. Are ⟷ CT and ⟷ DG parallel? Why?
10. Are ⟷ MP and ⟷ OA parallel? Why?
11. Consider the diagram shown.
a. Are ⟷ AB and ⟷ CD parallel? Why?
b. Assume ⟷ AB ∥ ⟷ CD . Are lines ⟷ CD and ⟷ GH parallel? Why?
c. Are ⟷ CD and ⟷ EF parallel? Why?
12. In the diagram, ⟷ AC is parallel to ⟷ DF . Write the unknown angle measures in the diagram.
13. In the diagram, what must be the x and y values to make ⟷ AB ∥ ⟷ CD and ⟷ BC ∥ ⟷ DE ? Explain.
Remember
For problems 14–17, write an equivalent expression.
2 3 (x + 6)
3 4 (x + 8) 16. 1 2 (x + 10)
2 5 (x + 25)
18. Consider figure CDEF and the following rigid motions.
• 90° counterclockwise rotation around the origin
• Reflection across the y-axis
a. Graph the image of figure CDEF under the sequence of rigid motions in the given order.
Label the image A.
b. Graph the image of figure CDEF under the sequence of rigid motions in the opposite order. Label the image B
c. Does the order matter when applying this sequence of rigid motions? Explain.
19. Write an equivalent expression to 7 3 · 7 4 with only one base.
1. Given parallel lines �� and �� with transversal ��, find the value of x.
Interior and Exterior
2. Identify a location for an exterior angle adjacent to ∠F.
3. Consider △HAT in the diagram.
a. Name the exterior angles of △ HAT.
b. If the measure of ∠HTA is 58° and the measure of ∠THA is 86°, find the measures of the following angles.
m∠HAT =
m∠GAT =
m∠OHA =
m∠CTH =
Remote Interior Angles
4. Find the value of x.
5. Find the value of y.
6. Write equations that represent the angle relationships shown in the diagram.
7. Use the relationship between an exterior angle of a triangle and the remote interior angles to find the value of x.
Find the Angle Measure
For problems 8–13, find the value of x in the diagram by using any of the angle relationships you have learned. Label any additional angle measures you use to find the value of x.
• defined exterior angle and remote interior angles of a triangle.
• determined that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle.
• solved equations to find angle measures.
Examples
1. Find the measure of ∠ ACD.
∠ ACB is adjacent to ∠ ACD, so ∠ ABC and ∠ BAC are the remote interior angles to ∠ ACD.
An exterior angle of a triangle is an angle that forms a linear pair with an interior angle of the triangle. In the diagram, ∠1 is an exterior angle of the triangle.
Remote interior angles of a triangle are the two interior angles not adjacent to a given exterior angle of the triangle. In the diagram, ∠1 is an exterior angle, and ∠2 and ∠3 are the remote interior angles.
∠ ACD is an exterior angle of △ ABC.
Sum of the remote interior angle measures Exterior angle measure
• determined the angle relationships in a given diagram.
• wrote equations by using angle relationships to find unknown values.
Examples
For problems 1 and 2, write an equation and find the value of x. Describe all angle relationships you use to write the equation.
x + 42 + 38 = 180 x + 80 = 180 x = 100
The value of x is 100.
The interior angle measures of a triangle sum to 180° .
Write an equation that shows the sum of all three interior angle measures equals 180° . Then solve for x
98 + 27 = x 125 = x
The sum of the measures of the remote interior angles equals the measure of the exterior angle, x° .
The value of x is 125.
The exterior angle measure of a triangle is equal to the sum of the remote interior angle measures.
3. Parallel lines �� and �� are cut by transversal ��. Write an equation and find the value of x. Identify all angle relationships you use to write the equation.
This angle measure is 62° because corresponding angles of parallel lines are equal in measure.
The value of x is 118.
62 + x = 180 x = 118
The measures of the corresponding angles are equal because lines �� and �� are parallel. Linear pairs have measures that sum to 180° .
For problems 1–3, write an equation by using angle relationships given in the diagram. Then find the measure of ∠1
For problems 4–6, write an equation by using angle relationships given in the diagram. Then find the value of x
x + 5)°
For problems 7 and 8, write equations by using angle relationships given in the diagram. Then find the value of x and y
9. Ava solves this problem correctly. Analyze Ava’s work. Then find the measure of ∠IFO by using a different strategy. Explain your work.
Ava’s Work:
+ 37 = y
= y
Because ↔ BI ∥ ⟷ AO , alternate interior angles ∠ BIT and ∠ FTO are congruent. So the measure of ∠FTO is 77°. The measure of the exterior angle of the triangle, ∠IFO, is equal to the sum of the measures of the remote interior angles, ∠ FTO and ∠TOF. So the measure of ∠IFO is 114° .
Remember
For problems 10–13, write an equivalent expression.
5(x + 3) + 8(x + 2)
14. Parallel lines �� and �� are cut by transversal ��.
a. In the diagram, ∠2 and ∠6 have what angle relationship?
b. Use rigid motions to describe how you know ∠2 and ∠6 are congruent.
15. Consider the equation 27 · 92 3n = 3 6. What is the value of n?
10.
11. 9(x − 5) + 3(x 2)
12. −2(x 4) + 4(x + 2)
13. 3(x − 4) − 7(x − 1)
Congruent Figures and the Pythagorean Theorem
Student Edition: Grade 8, Module 2, Topic D
Desire Paths
TOPIC D
Consider a right turn on a sidewalk. Have you ever walked across the grass to cut through a corner to save yourself a little time?
Of course you have. Who hasn’t?
That means you’ve benefitted, in a way, from the Pythagorean theorem. The two sidewalk edges you skipped and the grassy path you walked make a right triangle. If we measure the sidewalk edges, then we can use the Pythagorean theorem to determine the exact length of your shortcut.
These shortcuts are sometimes called “desire paths.” Look at where the grass is worn down and you can see where people desire to walk (rather than where the urban planner told them to walk by laying down pavement).
Often, what people desire—even without thinking about geometry—is a hypotenuse!
Given: A right triangle has leg lengths a and b and hypotenuse length c
Prove: a2 + b2 = c2
1. Use the figure to complete parts (a)–(h).
a. Label the side lengths and angle measures of the three unlabeled triangles.
b. Defend the statement: The entire figure is a square with side length a + b.
c. Can you conclude that the unshaded figure in the middle is a square? Why?
d. Consider the interior angles of one of the triangles. Defend the statement: x + y + 90 = 180.
e. What does the equation x + y + 90 = 180 tell you about the measures of the interior angles of the unshaded figure? Why?
f. Defend the statement: The unshaded figure is a square.
g. What is the area of the unshaded square?
h. Defend the statement: The area of the entire figure is the same as the combined areas of the unshaded square and all four triangles.
2. Use the figure to complete parts (a)–(f).
a. Label the side lengths of the three unlabeled triangles.
b. Is the figure on the bottom right a square? How do you know?
c. Is the figure on the top left a square? How do you know?
d. What is the area of the square on the bottom right?
e. What is the area of the square on the top left?
f. Defend the statement: The area of the entire figure is the same as the combined areas of the bottom right square, the top left square, and all four triangles.
3. Use the following figures to complete parts (a) and (b).
a. Write the areas of the outlined squares on the lines provided in the figures.
b. Explain how the two figures prove that a2 + b2 = c2 .
For problems 1–3, determine whether the statement is true or false. If the statement is false, explain your reasoning.
1. If I am 13 years old, then I am a teenager.
2. If I am a teenager, then I am 13 years old.
3. If Mr. Adams is Sara’s teacher, then Sara is Mr. Adams’s student.
4. Write the converse of the statement in problem 3. Then state whether the converse is true or false.
Converse of the Pythagorean Theorem
5. Consider this statement of the Pythagorean theorem for a triangle with side lengths a, b, and c, where c is the length of the longest side. If the triangle is a right triangle, then a2 + b2 = c 2 .
Write the converse of the Pythagorean theorem.
Proving the Converse
Given: A triangle has side lengths a, b, and c, where c is the length of the longest side, and a2 + b2 = c2 .
Prove: The triangle is a right triangle.
6. Use the following triangles to complete parts (a)–(d).
Triangle 1
a. Label the length of the hypotenuse x in triangle 2. Then use the Pythagorean theorem to write an equation to represent the relationship between the side lengths in triangle 2.
b. Defend the statement: Triangle 1 and triangle 2 have the same three side lengths.
c. Defend the statement: Triangle 1 and triangle 2 are congruent.
d. Finish the proof by explaining why triangle 1 must be a right triangle.
• found the length of a diagonal segment in the coordinate plane.
• formed right triangles in the coordinate plane so we could use the Pythagorean theorem to find the distance between two points.
Examples
1. Find the distance between points A and B.
Draw a segment connecting points A and B
Draw a right triangle with hypotenuse AB . Then count the units to find the leg lengths.
The length of AB is √ 164 units.
So the distance between points A and B is √ 164 units.
The length of AB is also the distance between points A and B
2. Is △ ABC a right triangle? Explain.
Because no side lengths of the triangle are on grid lines, draw segments that form right triangles for each pair of points. Then count the units to find the segment lengths.
Use the Pythagorean theorem three times to find the lengths of BC , AC , and AB .
Let x represent the length of BC :
3 2 + 3 2 = x 2
Let y represent the length of
z represent the length of
By the converse of the Pythagorean theorem, if (√ 18 ) 2 + (√ 8 ) 2 is equal to (√ 26 ) 2, then △ ABC is a right triangle. (√ 18 ) 2 + (√ 8 ) 2 = 18 + 8 = 26 (√ 26 ) 2 = 26
Because 26 = 26, we know (√ 18 ) 2 + (√ 8 ) 2 = (√ 26 ) 2 , so △ ABC is a right triangle.
Use the converse of the Pythagorean theorem: If the sum of the squares of the leg lengths equals the square of the hypotenuse length, then the triangle is a right triangle.
1. Dylan’s entertainment center has a rectangular opening for a TV. The opening is 50 inches wide and 30 inches tall. The size of a TV is described by the length of its diagonal.
Which is the largest TV from the list that Dylan can fit in his entertainment center? Explain.
Solving a Real-World Problem
2. In the book What’s Your Angle, Pythagoras?, Saltos and Pepros argue because their 12-foot ladder does not reach a temple roof, as shown. What height does their ladder reach on the temple wall? Round to the nearest tenth of a foot.
Solving a Mathematical Problem
3. The area of the right triangle is 26.46 square units. What is the perimeter of the triangle?
Maya wants to build a skateboard ramp with the dimensions shown. What length of plywood does Maya need for the top of the ramp? Round to the nearest tenth of an inch. 18 in
1. An 80-inch TV has a height of 39.2 inches. What is the width of the television? Round to the nearest tenth of an inch. 80 in
2. A 13-foot ladder leans on a wall as shown. What height does the ladder reach on the wall? Round to the nearest tenth of a foot.
3. Noor can take two paths from her house to Liam’s house. One path is direct. The other path requires Noor to travel on two different roads, as shown.
House
2 miles Noor’s House
miles
How much shorter is the direct path than the path with two different roads?
4. Consider the diagram of a portable soccer goal. The black lines show the frame of the goal.
How many feet of framing are needed for the frame of the goal? Round to the nearest tenth of a foot.
5. The area of the given right triangle is 66.5 square units.
a. What is the unknown leg length of the triangle?
b. What is the perimeter of the triangle? Round to the nearest tenth of a unit.
For problems 6 and 7, find the length of the diagonal c in the box shaped like a right rectangular prism. Round your final answer to the nearest tenth of an inch.
Remember
For problems 8–11, write an equivalent expression.
1. What did you assume about the amusement park situation to solve the problem in the lesson?
2. What tools did you use to solve the problem in the lesson?
3. Mr. Adams goes to the store to get milk and apples.
a. Sketch a path that takes Mr. Adams from the starting point, to both items, and then to the checkout.
b. If Mr. Adams walks at a rate of 3 feet per second, about how long will it take him to reach the checkout?
c. What did you assume to solve this problem?
Remember
For problems 4–7, write an equivalent expression.
8. Find the unknown side length.
9. Which rigid motions map a segment onto a segment of the same length? Choose all that apply.
A. Translation
B. Reflection
C. Rotation
Student Edition: Grade 8, Module 2, Mixed Practice 1
Mixed Practice 1
1. Henry created an equivalent expression for
by using the following steps. Ava thinks Henry made an error.
a. In which step, if any, did Henry make an error? Explain.
b. If Henry made an error, correct the error. Then find an equivalent expression.
For problems 2–5, solve and graph the solution to the inequality.
2. 3 x + 5 < 8
3. −4 x + 3 ≤ 7
4. 6 x + 4 ≥ 10
+ 2 > 7
5. −5 x
6. Consider the given diagram where ⟷ DB and ⟷ CF meet at point A. Point A is also the endpoint of ⟶ AE
a. Describe an angle relationship that would help you solve for x
b. Write an equation to find the value of x. Then determine the measure of ∠EAD.
7. Which expression is equivalent to 5 6 5 4 ?
A. 25 24
B. 10 10
C. 5 24
D. 5 10
8. Consider the equation 8 15 ⋅ 8 x = 8 30. Which equation can be used to determine the value of x ?
A. 15 + x = 30
B. 15 − x = 30
C. 15 ⋅ x = 30
D. 15 ÷ x = 30
9. While working on calculations for her science homework, Eve’s calculator displays the following: 4.1633363e -17
Write this number in scientific notation to help Eve interpret her calculator’s display.
10. Scientists believe that Jupiter is about 391 million miles away from Earth and that Mars is about 49 million miles away from Earth.
a. Approximate the distance from Jupiter to Earth as a single digit times a power of 10.
b. Approximate the distance from Mars to Earth as a single digit times a power of 10.
c. The distance from Jupiter to Earth is approximately how many times as far as the distance from Mars to Earth?
11. The total volume of fresh water on Earth is approximately 3.5 × 10 7 km 3. The total volume of all water on Earth is approximately 1.4 × 10 9 km 3. Of the approximate total volume of water on Earth, how much of it is not fresh water? Write your answer in scientific notation.
1. Rectangle 2 is shown. Rectangle 2 is a scale drawing of rectangle 1, which is not shown. The area of rectangle 1 is 81 square units. What scale factor is used to relate the side lengths of rectangle 1 to the side lengths of rectangle 2?
3. Find the side length of a cube that has a volume of 343 in3.
4. Using the information in the table, write an equation to show how the total cost relates to the number of containers of strawberries. Let t represent the total cost in dollars. Let s represent the number of containers of strawberries.
Rectangle 2
2. Solve x 2 = 196 for x.
5. Lily invests $3000 in a savings plan. The plan pays 1.25% simple interest at the end of each year. Lily does not make any deposits or withdrawals from the savings plan.
a. What is the balance of Lily’s savings plan at the end of 3 years?
b. Lily’s savings plan has a balance of at least $3200 at the end of how many years?
6. Eve buys 3 T-shirts at a store. She receives a 5% discount and pays an 8% sales tax. If each T-shirt costs $10, what is the total cost for the 3 T-shirts after the discount and sales tax are applied?
7. Indicate whether the decimal form of the number terminates or repeats.
8. Consider the following numbers:
a. Use approximation to place each number on the number line.
b. Classify each number as rational or irrational.
Student Edition: Grade 8, Module 2, Topic A, Lesson 3
1. Draw and label the image of △ ABC under a translation along ⟶
Student Edition: Grade 8, Module 2, Topic A, Lesson 4
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Adriana Akers, Amanda Aleksiak, Tiah Alphonso, Lisa Babcock, Christopher Barbee, Reshma P Bell, Chris Black, Erik Brandon, Beth Brown, Amanda H. Carter, Leah Childers, David Choukalas, Mary Christensen-Cooper, Cheri DeBusk, Jill Diniz, Mary Drayer, Karen Eckberg, Dane Ehlert, Samantha Falkner, Scott Farrar, Kelli Ferko, Krysta Gibbs, Winnie Gilbert, Danielle Goedel, Julie Grove, Marvin E. Harrell, Stefanie Hassan, Robert Hollister, Rachel Hylton, Travis Jones, Kathy Kehrli, Raena King, Emily Koesters, Liz Krisher, Alonso Llerena, Gabrielle Mathiesen, Maureen McNamara Jones, Pia Mohsen, Bruce Myers, Marya Myers, Kati O’Neill, Ben Orlin, April Picard, John Reynolds, Bonnie Sanders, Aly Schooley, Erika Silva, Hester Sofranko, Bridget Soumeillan, Ashley Spencer, Danielle Stantoznik, Tara Stewart, James Tanton, Cathy Terwilliger, Cody Waters, Valerie Weage, Allison Witcraft, Caroline Yang
Trevor Barnes, Brianna Bemel, Adam Cardais, Christina Cooper, Natasha Curtis, Jessica Dahl, Brandon Dawley, Delsena Draper, Sandy Engelman, Tamara Estrada, Soudea Forbes, Jen Forbus, Reba Frederics, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Nathan Hall, Cassie Hart, Marcela Hernandez, Rachel Hirsh, Abbi Hoerst, Libby Howard, Amy Kanjuka, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Crystal Love, Maya Márquez, Siena Mazero, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry, Mary-Lise Nazaire, Corinne Newbegin, Max Oosterbaan, Tamara Otto, Christine Palmtag, Andy Peterson, Lizette Porras, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Samuel Weyand, Dave White, Charmaine Whitman, Nicole Williams, Glenda Wisenburn-Burke, Howard Yaffe
Student Edition: Grade 8, Module 2, Talking Tool
Talking Tool
Share Your Thinking
I know . . . . I did it this way because . . . . The answer is because . . . . My drawing shows . . . .
Agree or Disagree
I agree because . . . . That is true because . . . . I disagree because . . . . That is not true because . . . .
Do you agree or disagree with ? Why?
Ask for Reasoning
Why did you . . . ? Can you explain . . . ? What can we do first? How is related to ?
Say It Again
I heard you say . . . . said . . . .
Another way to say that is . . . . What does that mean?
Thinking Tool
When I solve a problem or work on a task, I ask myself
Before
Have I done something like this before? What strategy will I use? Do I need any tools?
During Is my strategy working? Should I try something else? Does this make sense?
After
What worked well?
What will I do differently next time?
At the end of each class, I ask myself
What did I learn?
What do I have a question about?
MATH IS EVERYWHERE
Do you want to compare how fast you and your friends can run?
Or estimate how many bees are in a hive?
Or calculate your batting average?
Math lies behind so many of life’s wonders, puzzles, and plans.
From ancient times to today, we have used math to construct pyramids, sail the seas, build skyscrapers—and even send spacecraft to Mars.
Fueled by your curiosity to understand the world, math will propel you down any path you choose.
Ready to get started?
Module 1
Scientific Notation, Exponents, and Irrational Numbers
Module 2
Rigid Motions and Congruent Figures
Module 3
Dilations and Similar Figures
Module 4
Linear Equations in One and Two Variables
Module 5
Systems of Linear Equations
Module 6
Functions and Bivariate Statistics
What does this painting have to do with math?
Abstract expressionist Al Held was an American painter best known for his “hard edge” geometric paintings. His bright palettes and bold forms create a three-dimensional space that appears to have infinite depth. Held, who sometimes found inspiration in architecture, would often play with the viewer’s sense of visual perception. While most of Held’s artworks are paintings, he also worked in mosaic and stained glass.
