1. Splash Screen

2. Five-Minute Check (over Lesson 7–2) CCSS Then/Now
Postulate 7.1: Angle-Angle (AA) Similarity
Example 1: Use the AA Similarity Postulate
Theorems
Proof: Theorem 7.2
Example 2: Use the SSS and SAS Similarity Theorems
Example 3: Standardized Test Example: Sufficient Conditions
Theorem 7.4: Properties of Similarity
Example 4: Parts of Similar Triangles
Example 5: Real-World Example: Indirect Measurement
Concept Summary: Triangle Similarity
Lesson Menu

3. Determine whether the triangles are similar.
A. Yes, corresponding angles are congruent and corresponding sides are proportional.
B. No, corresponding sides are not proportional.
5-Minute Check 1

4. Determine whether the triangles are similar.
A. Yes, corresponding angles are congruent and corresponding sides are proportional.
B. No, corresponding sides are not proportional.
5-Minute Check 1

5. The quadrilaterals are similar
The quadrilaterals are similar. Find the scale factor of the larger quadrilateral to the smaller quadrilateral.
A. 5:3
B. 4:3
C. 3:2
D. 2:1
5-Minute Check 2

6. The quadrilaterals are similar
The quadrilaterals are similar. Find the scale factor of the larger quadrilateral to the smaller quadrilateral.
A. 5:3
B. 4:3
C. 3:2
D. 2:1
5-Minute Check 2

7. The triangles are similar. Find x and y.
A. x = 5.5, y = 12.9
B. x = 8.5, y = 9.5
C. x = 5, y = 7.5
D. x = 9.5, y = 8.5
5-Minute Check 3

8. The triangles are similar. Find x and y.
A. x = 5.5, y = 12.9
B. x = 8.5, y = 9.5
C. x = 5, y = 7.5
D. x = 9.5, y = 8.5
5-Minute Check 3

9. __
Two pentagons are similar with a scale factor of The perimeter of the larger pentagon is 42 feet. What is the perimeter of the smaller pentagon? 3 7
A. 12 ft
B. 14 ft
C. 16 ft
D. 18 ft
5-Minute Check 4

10. __
Two pentagons are similar with a scale factor of The perimeter of the larger pentagon is 42 feet. What is the perimeter of the smaller pentagon? 3 7
A. 12 ft
B. 14 ft
C. 16 ft
D. 18 ft
5-Minute Check 4

11. G.SRT.4 Prove theorems about triangles.
Content Standards
G.SRT.4 Prove theorems about triangles.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Mathematical Practices
4 Model with mathematics.
7 Look for and make use of structure.
CCSS

12. Use similar triangles to solve problems.
You used the AAS, SSS, and SAS Congruence Theorems to prove triangles congruent.
Identify similar triangles using the AA Similarity Postulate and the SSS and SAS Similarity Theorems.
Use similar triangles to solve problems.
Then/Now

13. Concept

14. Use the AA Similarity Postulate
A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
Example 1

15. By the Triangle Sum Theorem, 42 + 58 + mA = 180, so mA = 80.
Use the AA Similarity Postulate
Since mB = mD, B D.
By the Triangle Sum Theorem, mA = 180, so mA = 80.
Since mE = 80, A E.
Answer:
Example 1

16. By the Triangle Sum Theorem, 42 + 58 + mA = 180, so mA = 80.
Use the AA Similarity Postulate
Since mB = mD, B D.
By the Triangle Sum Theorem, mA = 180, so mA = 80.
Since mE = 80, A E.
Answer: So, ΔABC ~ ΔEDF by the AA Similarity.
Example 1

17. Use the AA Similarity Postulate
B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
Example 1

18. QXP NXM by the Vertical Angles Theorem.
Use the AA Similarity Postulate
QXP NXM by the Vertical Angles Theorem.
Since QP || MN, Q N.
Answer:
Example 1

19. QXP NXM by the Vertical Angles Theorem.
Use the AA Similarity Postulate
QXP NXM by the Vertical Angles Theorem.
Since QP || MN, Q N.
Answer: So, ΔQXP ~ ΔNXM by AA Similarity.
Example 1

20. D. No; the triangles are not similar.
A. Determine whether the triangles are similar. If so, write a similarity statement.
A. Yes; ΔABC ~ ΔFGH
B. Yes; ΔABC ~ ΔGFH
C. Yes; ΔABC ~ ΔHFG
D. No; the triangles are not similar.
Example 1

21. D. No; the triangles are not similar.
A. Determine whether the triangles are similar. If so, write a similarity statement.
A. Yes; ΔABC ~ ΔFGH
B. Yes; ΔABC ~ ΔGFH
C. Yes; ΔABC ~ ΔHFG
D. No; the triangles are not similar.
Example 1

22. D. No; the triangles are not similar.
B. Determine whether the triangles are similar. If so, write a similarity statement.
A. Yes; ΔWVZ ~ ΔYVX
B. Yes; ΔWVZ ~ ΔXVY
C. Yes; ΔWVZ ~ ΔXYV
D. No; the triangles are not similar.
Example 1

23. D. No; the triangles are not similar.
B. Determine whether the triangles are similar. If so, write a similarity statement.
A. Yes; ΔWVZ ~ ΔYVX
B. Yes; ΔWVZ ~ ΔXVY
C. Yes; ΔWVZ ~ ΔXYV
D. No; the triangles are not similar.
Example 1

24. Concept

25. Concept

26. Use the SSS and SAS Similarity Theorems
A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
Answer:
Example 2

27. Answer: So, ΔABC ~ ΔDEC by the SSS Similarity Theorem.
Use the SSS and SAS Similarity Theorems
A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
Answer: So, ΔABC ~ ΔDEC by the SSS Similarity Theorem.
Example 2

28. By the Reflexive Property, M  M.
Use the SSS and SAS Similarity Theorems
B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
By the Reflexive Property, M  M.
Answer:
Example 2

29. By the Reflexive Property, M  M.
Use the SSS and SAS Similarity Theorems
B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
By the Reflexive Property, M  M.
Answer: Since the lengths of the sides that include M are proportional, ΔMNP ~ ΔMRS by the SAS Similarity Theorem.
Example 2

30. A. ΔPQR ~ ΔSTR by SSS Similarity Theorem
A. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.
A. ΔPQR ~ ΔSTR by SSS Similarity Theorem
B. ΔPQR ~ ΔSTR by SAS Similarity Theorem
C. ΔPQR ~ ΔSTR by AA Similarity Theorem
D. The triangles are not similar.
Example 2

31. A. ΔPQR ~ ΔSTR by SSS Similarity Theorem
A. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.
A. ΔPQR ~ ΔSTR by SSS Similarity Theorem
B. ΔPQR ~ ΔSTR by SAS Similarity Theorem
C. ΔPQR ~ ΔSTR by AA Similarity Theorem
D. The triangles are not similar.
Example 2

32. A. ΔAFE ~ ΔABC by SAS Similarity Theorem
B. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.
A. ΔAFE ~ ΔABC by SAS Similarity Theorem
B. ΔAFE ~ ΔABC by SSS Similarity Theorem
C. ΔAFE ~ ΔACB by SAS Similarity Theorem
D. ΔAFE ~ ΔACB by SSS Similarity Theorem
Example 2

33. A. ΔAFE ~ ΔABC by SAS Similarity Theorem
B. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.
A. ΔAFE ~ ΔABC by SAS Similarity Theorem
B. ΔAFE ~ ΔABC by SSS Similarity Theorem
C. ΔAFE ~ ΔACB by SAS Similarity Theorem
D. ΔAFE ~ ΔACB by SSS Similarity Theorem
Example 2

34. Sufficient Conditions
If ΔRST and ΔXYZ are two triangles such that = , which of the following would be sufficient to prove that the triangles are similar? A B C R  S D __ 2 3
___ RS XY
Example 3

35. Sufficient Conditions
Read the Test Item
You are given that = and asked to identify which additional information would be sufficient to prove that ΔRST ~ ΔXYZ. __ 2 3
___ RS XY
Example 3

36. Sufficient Conditions__ 2 3
Solve the Test Item
Since = , you know that these two sides are proportional with a scale factor of Check each answer choice until you find one that supplies sufficient information to prove that ΔRST ~ ΔXYZ.
___ RS XY
Example 3

37. Sufficient Conditions__ 2 3
Choice A
If = , then you know that the other two sides are proportional. You do not, however, know whether the scale factor is , as determined by Therefore, this is not sufficient information.
___ RT XZ ST YZ RS XY
Example 3

38. Sufficient Conditions__ 2 3
Choice B
If = = , then you know that all the sides are proportional with the same scale factor, This is sufficient information by the SSS Similarity Theorem to determine that the triangles are similar.
___ RS XY RT XZ
Example 3

39. Sufficient Conditions__ 2 3
Choice B
If = = , then you know that all the sides are proportional with the same scale factor, This is sufficient information by the SSS Similarity Theorem to determine that the triangles are similar.
___ RS XY RT XZ
Answer:
Example 3

40. Sufficient Conditions__ 2 3
Choice B
If = = , then you know that all the sides are proportional with the same scale factor, This is sufficient information by the SSS Similarity Theorem to determine that the triangles are similar.
___ RS XY RT XZ
Answer: B
Example 3

41. A. = B. mA = 2mD C. = D. = AC DC 4 3 BC 5 EC
Given ΔABC and ΔDEC, which of the following would be sufficient information to prove the triangles are similar?
A =
B. mA = 2mD
C =
D =
___ AC DC __ 4 3 BC 5 EC
Example 3

42. A. = B. mA = 2mD C. = D. = AC DC 4 3 BC 5 EC
Given ΔABC and ΔDEC, which of the following would be sufficient information to prove the triangles are similar?
A =
B. mA = 2mD
C =
D =
___ AC DC __ 4 3 BC 5 EC
Example 3

43. Concept

44. Parts of Similar Triangles
ALGEBRA Given , RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT.
Example 4

45. Cross Products Property
Parts of Similar Triangles
Since because they are alternate interior angles. By AA Similarity, ΔRSQ ~ ΔTUQ. Using the definition of similar polygons,
Substitution
Cross Products Property
Example 4

46. Distributive Property
Parts of Similar Triangles
Distributive Property
Subtract 8x and 30 from each side.
Divide each side by 2.
Now find RQ and QT.
Answer:
Example 4

47. Distributive Property
Parts of Similar Triangles
Distributive Property
Subtract 8x and 30 from each side.
Divide each side by 2.
Now find RQ and QT.
Answer: RQ = 8; QT = 20
Example 4

48. ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC.
Example 4

49. ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC.
Example 4

50. Understand Make a sketch of the situation.
Indirect Measurement
SKYSCRAPERS Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 p.m. The length of the shadow was 2 feet. Then he measured the length of Sears Tower’s shadow and it was 242 feet at the same time. What is the height of the Sears Tower?
Understand Make a sketch of the situation.
Example 5

51. So the following proportion can be written.
Indirect Measurement
Plan In shadow problems, you can assume that the angles formed by the Sun’s rays with any two objects are congruent and that the two objects form the sides of two right triangles. Since two pairs of angles are congruent, the right triangles are similar by the AA Similarity Postulate.
So the following proportion can be written.
Example 5

52. Cross Products Property
Indirect Measurement
Solve Substitute the known values and let x be the height of the Sears Tower.
Substitution
Cross Products Property
Simplify.
Divide each side by 2.
Example 5

53. Indirect Measurement
Answer:
Example 5

54. Answer: The Sears Tower is 1452 feet tall.
Indirect Measurement
Answer: The Sears Tower is 1452 feet tall.
Check The shadow length of the Sears Tower is or 121 times the shadow length of the light pole. Check to see that the height of the Sears Tower is 121 times the height of the light pole = 121 
______
242 2
1452 12
Example 5

55. LIGHTHOUSES On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 foot 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot?
A. 196 ft B ft
C. 441 ft D ft
Example 5

56. LIGHTHOUSES On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 foot 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot?
A. 196 ft B ft
C. 441 ft D ft
Example 5

57. Concept

58. End of the Lesson