1. Showing Triangles are Similar: AA, SSS and SAS
Sections 7.3 &7.4

2. Objectives
Identify similar triangles using the AA Similarity Postulate and the SSS and SAS Similarity Theorems (similarity shortcuts).
Use similar triangles to solve problems.

3. Key Vocabulary
None

4. Postulates
15 Angle – Angle (AA) Similarity Postulate

5. Theorems 7.2 Side – Side – Side (SSS) Similarity Theorem
7.3 Side – Angle – Side (SAS) Similarity Theorem

6. Do you really have to check all the sides and angles?
Similar Triangles
Are the triangles below similar?
Do you really have to check all the sides and angles?
NO, there are Shortcuts for determining Similarity.

7. Triangle Similarity Shortcuts

8. Postulate 15 Angle-Angle (AA) Similarity
Angle – Angle (AA) Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
Example: If ∠K≅∠Y and ∠J≅∠X, then ∆JKL∼∆XYZ.

9. Example 1a
A. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

10. Example 1a 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.

11. Example 1b
B. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

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

13. Your Turn:
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.

14. Your Turn:
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.

15. Thales
The Greek mathematician Thales was the first to measure the height of a pyramid by using geometry. He showed that the ratio of a pyramid to a staff was equal to the ratio of one shadow to another.

16. Example 2
If the shadow of the pyramid is 576 feet, the shadow of the staff is 6 feet, and the height of the staff is 5 feet.
Explain why Thales’ method worked to find the height of the pyramid?
Find the height of the pyramid?

17. Example 2
Explain why Thales’ method worked to find the height of the pyramid?
The Triangles formed by the pyramid and its shadow, and the staff and its shadow are similar triangles by the AA Postulate.
Find the height of the pyramid?
Solution: The pyramid is 480 ft tall.
Similarity proportion
Substitution
Cross product
Simplify
Divide by 6

18. If two pairs of angles are congruent, then the triangles are similar.
Example 1
Use the AA Similarity Postulate
Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning.
SOLUTION
If two pairs of angles are congruent, then the triangles are similar. 1.
G  L because they are both marked as right angles.

19. Find mF to determine whether F is congruent to J.
Example 1
Use the AA Similarity Postulate 2.
Find mF to determine whether F is congruent to J.
mF + 90° + 61° = 180°
Triangle Sum Theorem
mF ° = 180°
Add.
mF = 29°
Subtract 151° from each side.
Both F and J measure 29°, so F  J.
ANSWER
By the AA Similarity Postulate, FGH ~ JLK

20. Are you given enough information to show that RST is similar to
Example 2
Use the AA Similarity Postulate
Are you given enough information to show that RST is similar to
RUV? Explain your reasoning.
SOLUTION
Redraw the diagram as two triangles: RUV and RST.
From the diagram, you know that both RST and RUV measure 48°, so RST  RUV. Also, R  R by the Reflexive Property of Congruence. By the AA Similarity Postulate, RST ~ RUV.

21. Your Turn
Use the AA Similarity Postulate
Determine whether the triangles are similar. If they are similar, write a similarity statement. 1.
ANSWER
yes; RST ~ MNL 2.
ANSWER
yes; GLH ~ GKJ

22. Use Similar Triangles Your Turn
Write a similarity statement for the triangles. Then find the value of the variable. 3.
ANSWER
ABC ~ DEF; 9 4.
ABD ~ EBC; 3
ANSWER

23. Assignment 7.3
Pg 375 – 378: #1 – 45 odd

24. Theorem 7.2 Side-Side-Side (SSS) Similarity
Side – Side – Side (SSS) Similarity Theorem:
If the corresponding side lengths of two triangles are proportional, then the two triangles are similar.
Example:

25. Theorem 7.3 Side-Angle-Side (SAS) Similarity
Side – Angle Side (SAS) Similarity Theorem:
If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the two triangles are similar.
Example:

26. Review - Similar Triangles
Previously, we learned how to determine if two triangles were congruent (SSS, SAS, ASA, AAS). There are also several tests to prove triangles are similar.
Postulate 15 – AA Similarity 2 s of a Δ are  to 2 s of another Δ
Theorem 7.2 – SSS Similarity 3 corresponding sides of 2 Δs are proportional
Theorem 7.3 – SAS Similarity 2 corresponding sides of 2 Δs are proportional and the included s are 

27. Example 3a
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.

28. Example 3b
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.

29. Your Turn:
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 AAA Similarity Theorem
D. The triangles are not similar.

30. Example 3b
B. Determine whether the triangles are similar. If so, choose the correct similarity statement to match the given data.
A. The triangles are not similar.
B. ΔAFE ~ ΔACB by SSS Similarity Theorem
C. ΔAFE ~ ΔAFC by SSS Similarity Theorem
D. ΔAFE ~ ΔBCA by SSS Similarity Theorem

31. Example 4:
In the figure, and Determine which triangles in the figure are similar.

32. Example 4: by the Alternate Interior Angles Theorem.
Vertical angles are congruent,
Answer: Therefore, by the AA Similarity Theorem,

33. Your Turn:
In the figure, OW = 7, BW = 9, WT = 17.5, and WI = Determine which triangles in the figure are similar. I
Answer:

34. Example 5:
ALGEBRA Given QT 2x 10, UT 10, find RQ and QT.

35. Example 5:
Since because they are alternate interior angles. By AA Similarity, Using the definition of similar polygons,
Substitution
Cross products

36. Example 5: Distributive Property Subtract 8x and 30 from each side.
Divide each side by 2.
Now find RQ and QT.
Answer:

37. Your Turn:
ALGEBRA Given and CE x + 2, find AC and CE.
Answer:

38. Example 6
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

39. Example 6 __ 2 3
If = = , then you know that all the sides are proportional by 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

40. Your Turn: 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

41. Example 7:
INDIRECT MEASUREMENT: 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 the Sears Tower’s shadow and it was 242 feet at that time. What is the height of the Sears Tower?

42. Example 7:
Assuming that the sun’s rays form similar triangles, the following proportion can be written.
Now substitute the known values and let x be the height of the Sears Tower.
Substitution
Cross products

43. Example 7: Simplify. Divide each side by 2.
Answer: The Sears Tower is 1452 feet tall.

44. Your Turn:
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 feet 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?
Answer: 196 ft

45. Find the ratios of the corresponding sides. SOLUTION
Example 1
Use the SSS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A.
Find the ratios of the corresponding sides.
SOLUTION
All three ratios are equal. So, the corresponding sides of the triangles are proportional. PR SU 12 6
12 ÷ 6
6 ÷ 6 = 2 1 RQ UT 10 5
10 ÷ 5
5 ÷ 5 QP TS 8 4
8 ÷ 4
4 ÷ 4 45

46. The scale factor of Triangle B to Triangle A is . 2 1
Example 1
Use the SSS Similarity Theorem
ANSWER
The scale factor of Triangle B to Triangle A is . 2 1
By the SSS Similarity Theorem, PQR ~ STU. 46

47. Is either DEF or GHJ similar to ABC?
Example 2
Use the SSS Similarity Theorem
Is either DEF or GHJ similar to ABC?
SOLUTION
Look at the ratios of corresponding sides in ABC and DEF. 1.
Shortest sides = 6 4 AB DE 3 2
Longest sides 12 8 CA FD
Remaining sides 9 BC EF
ANSWER
Because all of the ratios are equal, ABC ~ DEF. 47

48. Look at the ratios of corresponding sides in ABC and GHJ. 2.
Example 2
Use the SSS Similarity Theorem
Look at the ratios of corresponding sides in ABC and GHJ. 2.
Shortest sides = 6 AB GH 1
Longest sides 12 14 CA JG 7
Remaining sides BC HJ 9 10
ANSWER
Because the ratios are not equal, ABC and GHJ are not similar. 48

49. Your Turn
Use the SSS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement. 1.
ANSWER
yes; ABC ~ DFE 2.
ANSWER no

50. C and F both measure 61°, so C  F.
Example 3
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement.
SOLUTION
C and F both measure 61°, so C  F.
Compare the ratios of the side lengths that include C and F. = AC DF 3 5
Shorter sides 6 10 CB FE
Longer sides
The lengths of the sides that include C and F are proportional.
ANSWER
By the SAS Similarity Theorem, ABC ~ DEF. 50

51. Separate the triangles, VYZ and VWX, and label the side lengths.
Example 4
Similarity in Overlapping Triangles
Show that VYZ ~ VWX.
Separate the triangles, VYZ and VWX, and label the side lengths.
SOLUTION
V  V by the Reflexive Property of Congruence.
Shorter sides
4 + 8 4 = VY VW 12 3 1
Longer sides
5 + 10 5 ZV XV 15 51

52. The lengths of the sides that include V are proportional.
Example 4
Similarity in Overlapping Triangles
The lengths of the sides that include V are proportional.
ANSWER
By the SAS Similarity Theorem, VYZ ~ VWX. 52

53. Your Turn
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 3.
ANSWER
No; H  M but 6 8 12 ≠ .

54. lengths of the sides that include P are proportional,
Your Turn
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 4.
ANSWER
Yes; P  P, and the
lengths of the sides that include P are proportional,
so PQR ~ PST by the SAS Similarity Theorem. , 6 3 PS PQ = 2 1 10 5 PT PR ;

55. Summary of Similarity Shortcuts

56. Assignment 7.4
Pg. 382 – 385: # 1 – 29 odd, 33 – 37 odd

57. Joke Time Why was the elephant standing on the marshmallow?
He didn’t want to fall in the hot chocolate!
What would you say if someone took your playing cards?
dec-a-gon
What kind of insect is good at math?
An account-ant