1. 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

2. 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

3. Concept

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

5. 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

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

7. 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

8. 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

9. 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

10. Concept

11. Concept

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. Concept

22. Concept

23. Find the geometric mean between 2 and 50.
Let x represent the geometric mean.
Definition of geometric mean
Cross products
Take the positive square root of each side.
Simplify.
Answer: The geometric mean is 10.
Example 1

24. A. Find the geometric mean between 3 and 12.
Example 1

25. Concept

26. Separate the triangles into two triangles along the altitude.
Identify Similar Right Triangles
Write a similarity statement identifying the three similar triangles in the figure.
Separate the triangles into two triangles along the altitude.
Example 2

27. Answer: So, by Theorem 8.1, ΔEGF ~ ΔFGH ~ ΔEFH.
Identify Similar Right Triangles
Then sketch the three triangles, reorienting the smaller ones so that their corresponding angles and sides are in the same position as the original triangle.
Answer: So, by Theorem 8.1, ΔEGF ~ ΔFGH ~ ΔEFH.
Example 2

28. Write a similarity statement identifying the three similar triangles in the figure.
A. ΔLNM ~ ΔMLO ~ ΔNMO
B. ΔNML ~ ΔLOM ~ ΔMNO
C. ΔLMN ~ ΔLOM ~ ΔMON
D. ΔLMN ~ ΔLMO ~ ΔMNO
Example 2

29. Concept

30. Use Geometric Mean with Right Triangles
Find c, d, and e.
Example 3

31. Geometric Mean (Altitude) Theorem
Use Geometric Mean with Right Triangles
Since e is the measure of the altitude drawn to the hypotenuse of right ΔJKL, e is the geometric mean of the lengths of the two segments that make up the hypotenuse, JM and ML.
Geometric Mean (Altitude) Theorem
Substitution
Simplify.
Example 3

32. Geometric Mean (Leg) Theorem
Use Geometric Mean with Right Triangles
Since d is the measure of leg JK, d is the geometric mean of JM, the measure of the segment adjacent to this leg, and the measure of the hypotenuse JL.
Geometric Mean (Leg) Theorem
Substitution
Use a calculator to simplify.
Example 3

33. Geometric Mean (Leg) Theorem
Use Geometric Mean with Right Triangles
Since c is the measure of leg KL, c is the geometric mean of ML, the measure of the segment adjacent to KL, and the measure of the hypotenuse JL.
Geometric Mean (Leg) Theorem
Substitution
Use a calculator to simplify.
Answer: e = 12, d ≈ 13.4, c ≈ 26.8
Example 3

34. Find e to the nearest tenth.
B. 24
C. 17.9
D. 11.3
Example 3