Proving Triangles Similar GEOMETRY LESSON 8-3

(For help, go to Lessons 4-2 and 4-3.)

Name the postulate or theorem you can use to prove the triangles congruent.

1.

3.

2.

Proving Triangles Similar GEOMETRY LESSON 8-3

Solutions 1. All corresponding sides are marked congruent, so Side-Side-Side, or SSS postulate 2. Two sides and an included angle of one are congruent to two sides and an included angle of the other, so Side-Angle-Side, or SAS Postulate 3. Two angles and an included side of one are congruent to two angles and an included side of the other, so Angle-Side-Angle, or ASA Postulate

Proving Triangles Similar GEOMETRY LESSON 8-3

MX AB. Explain why the triangles are similar. Write a similarity statement.

Because MX AB, AXM BXK. A

AXM and

BXK are both right angles, so

B because their measures are equal.

AMX ~ BKX by the Angle-Angle Similarity Postulate (AA ~ Postulate).

Proving Triangles Similar GEOMETRY LESSON 8-3

Explain why the triangles must be similar. Write a similarity statement.

YVZ

WVX because they are vertical angles.

VY 12 1 VZ 18 1 = = and = = , so corresponding sides are proportional. VW 24 2 VX 36 2

Therefore, YVZ ~ WVX by the Side-Angle-Side Similarity Theorem (SAS Similarity Theorem).

Proving Triangles Similar GEOMETRY LESSON 8-3

ABCD is a parallelogram. Find WY. Because ABCD is a parallelogram, AB || DC. XAW ZYW and AXW YZW because parallel lines cut by a transversal form congruent alternate interior angles. Therefore,

AWX ~

YWZ by the AA ~ Postulate.

Use the properties of similar triangles to find WY. WZ WY = WA WX 10 WY = 4 5 10 WY = 4 5

WY = 12.5

Proving Triangles Similar GEOMETRY LESSON 8-3

Joan places a mirror 24 ft from the base of a tree. When she stands 3 ft from the mirror, she can see the top of the tree reflected in it. If her eyes are 5 ft above the ground, how tall is the tree? Draw the situation described by the example. TR represents the height of the tree, point M represents the mirror, and point J represents Joanâ€™s eyes. Both Joan and the tree are perpendicular to the ground, so m JOM = m TRM, and therefore JOM TRM. The light reflects off a mirror at the same angle at which it hits the mirror, so JMO TMR. Use similar triangles to find the height of the tree.

Proving Triangles Similar GEOMETRY LESSON 8-3

(continued)

JOM ~ RM TR = OM JO TR = 24 3 5 24 TR = 3 5

TRM

5

TR = 40 The tree is 40 ft tall.

Proving Triangles Similar GEOMETRY LESSON 8-3

Pages 435-438 Exercises 1. Yes; ABC ~ SSS ~ Thm.

FED;

2. No; more info. is needed. 2

3. Ex. 1: (for ABC to 3 FED); Ex. 2: Not possible; the s arenâ€™t necessarily similar.

4. yes; FHG ~ AA ~ Post.

KHJ;

5. No;

6 10 . = / 3 4

11. AA ~ Post.; 2.5

6. No; 20 =/ 25 .

12. AA ~ Post.; 12 5

7. Yes; APJ ~ ABC; SSS ~ Thm. or SAS ~ Thm.

13. AA ~ Post.; 12

8. Yes; NMP ~ NQR; SAS ~ Thm.

15. AA , Post.; 15

45

9.

55

32 45 No; . =/ 22 30

6

14. AA ~ Post.; 8

16. SAS ~ Thm.; 12 m 17. AA ~ Post.; 220 yd

10. AA ~ Post.; 7.5

18. AA ~ Post.; 15 ft 9 in.

Proving Triangles Similar GEOMETRY LESSON 8-3

19. AA ~ Post.; 90 ft 20. Answers may vary. Sample: She can measure her shadow and use ~ s to find the length of the shadow of the proposed building. 21. 151 m 22. a. trapezoid

b.

RSZ ~ TWZ; AA ~ Post.

23. a. No; the corr. s may not be . b. Yes; every isosc. rt. is a 45°-45°90° . Therefore, by AA ~ Thm. they are all ~.

27. No; there is only one of each . 28. 45 ft

29. Check students’ work. 30. 3 : 2

24. Yes; GMK ~ SMP; SAS ~ Thm. 25. Yes; AWV ~ AST; SAS ~ Thm.

31. 2 : 1 32. 12 : 7

33. 4 : 3 26. Yes; XYZ ~ SSS ~ Thm.

MNK; 34. 3 : 1

Proving Triangles Similar GEOMETRY LESSON 8-3

35. 3 : 2

41. a. 98 m; 98 m

36. 3 : 2

b. 420 m2; 420 m2

37. 2 : 1

c. No; the s given are a counterexample to this conjecture, since the sides are not in proportion.

38. 3 : 1 39. 6 : 1 40. Check studentsâ€™ work. Draw ABC. Construct A R. Construct RS such that RS = 3AB, and RT such that RT = 3AC. Connect points S and T.

42. 1.

|| 2, EF AF, BC AF (Given)

2.

EFD and BCA are right s . (Def. of )

1

42. (continued) 3. EFD BCA (All rt. s are .) 4.

BAC EDF (If || lines, then corr. s are .)

5.

ABC ~ (AA ~)

6.

BC EF = AC DF

DEF

(Def. of similar)

Proving Triangles Similar GEOMETRY LESSON 8-3

43. 1.

BC EF = , EF AF, AC DF

BC 2.

3.

4.

5.

AF (Given)

ACB and are rt. s . (Def. of ) ACB (All rt.

s

DFE

45. C

,

44. 1. RT • TQ = MT • TS (Given)

DFE are .)

ABC ~ (SAS ~)

43. (continued) 6. 1 || 2, (If corr. s are then || lines.)

2. MT = RT (Prop. of TQ TS Proportions)

DEF

BAC EDF (Def. of similar)

3.

RTM STQ (Vert. s are .)

4.

RTM ~ STQ (SAS ~ Thm.)

46. I 47. [2] a. All corr. s of similar s are . Q X; subtract m V and m L from 180 to get m Q = m X. b. 52 [1] incorrect explanation OR incorrect measure

Proving Triangles Similar GEOMETRY LESSON 8-3

48. [4] a.

48. (continued) [2] correct diagram and similarity statement OR correct proportion [1] correct height, no work shown

ABC ~ ADE; AA , Post. 6 h b. = ; 18 ft 10 30

[3] appropriate methods and appropriate diagram but incorrect solution

49.

E

50.

P

53. EZ

54. YZ 55. x-values may vary. Sample: W(–b, c); Z(–b, –c) 56. x-values may vary. Sample: W(–b, c); Z(–a, 0) 57. 6 < x < 24

51.

Y

52. ZY

Proving Triangles Similar GEOMETRY LESSON 8-3

Are the triangles similar? If so, write a similarity statement and name the postulate or theorem you used. If not, explain. The congruent angle is not included between 1. 2. the proportional sides, so you cannot conclude that the triangles are similar. ABX ~ CDX by SAS ~ Theorem

3.

4. Find the value of x. 16

MRT ~

XRW by AA ~ Postulate

5. When a 6 ft tall man casts a shadow 18 ft long, a nearby tree casts a shadow 93 ft long. How tall is the tree? 31 ft