Warm Up

1. Prove Triangles Similar by AA Warm Up Lesson Presentation Lesson Quiz

2. 1.InABCandXZW,mA =mXand mB=mZ.What can you conclude aboutmCandmW? They are the same. ANSWER 54 x 2.Solve = . 9 18 108 ANSWER Warm-Up

3. 3.ABCDEF.Findx. ~ 10 ANSWER Warm-Up

4. Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. Example 1

5. Because they are both right angles, Dand Gare congruent. By the Triangle Sum Theorem, 26° + 90° +m E= 180°, so m E= 64°. Therefore, Eand Hare congruent. Example 1 SOLUTION So, ∆CDE~∆KGHby the AA Similarity Postulate.

6. a. ∆ABEand ∆ACD a. You may find it helpful to redraw the triangles separately. Because mABE and mC both equal 52°,ABEC.By the Reflexive Property, AA. Example 2 Show that the two triangles are similar. SOLUTION So, ∆ ABE~ ∆ ACDby the AA Similarity Postulate.

7. b. ∆SVRand ∆UVT b. You know SVR UVTby the Vertical Angles Congruence Theorem. The diagram shows RS||UTso S Uby the Alternate Interior Angles Theorem. Example 2 Show that the two triangles are similar. SOLUTION So, ∆SVR~ ∆UVTby the AA Similarity Postulate.

8. 1. ∆FGHand ∆RQS ANSWER In each triangle all three angles measure 60°, so by the AA similarity postulate, the triangles are similar ∆FGH ~ ∆QRS. Guided Practice Show that the triangles are similar. Write a similarity statement.

9. 2. ∆CDFand ∆DEF ANSWER Since m CDF= 58° by the Triangle Sum Theorem and mDFE= 90° by the Linear Pair Postulate the two triangles are similar by theAASimilarity Postulate; ∆CDF~ ∆DEF. Guided Practice Show that the triangles are similar. Write a similarity statement.

10. 3. REASONING Suppose in Example 2, part (b),SRTU. Could the triangles still be similar? Explain. ANSWER Yes; if S T, the triangles are similar by the AA Similarity Postulate. Guided Practice

11. Example 3

12. Example 3 SOLUTION The flagpole and the woman form sides of two right triangles with the ground, as shown below. The sun’s rays hit the flagpole and the woman at the same angle. You have two pairs of congruent angles, so the triangles are similar by the AA Similarity Postulate.

13. 50 ft x ft = 40 in. 64 in. ANSWER The flagpole is 80 feet tall. The correct answer is C. Example 3 You can use a proportion to find the height x. Write 5 feet 4 inches as 64 inches so that you can form two ratios of feet to inches. Write proportion of side lengths. 40x = 64(50) Cross Products Property x = 80 Solve for x.

14. 4. WHAT IF? Achild who is 58 inches tall is standing next to the woman in Example 3. How long is the child’s shadow? 36.25 in. ANSWER Guided Practice

15. 5. You are standing in your backyard, and you measure the lengths of the shadows cast by both you and a tree. Write a proportion showing how you could find the height of the tree. SAMPLE ANSWER length of shadow tree height = length of your shadow your height Guided Practice

16. 1. ANSWER Yes; ABE ~ ACD Lesson Quiz Determine if the two triangles are similar. If they are write a similarity statement.

17. 2. no ANSWER Lesson Quiz Determine if the two triangles are similar. If they are write a similarity statement.

18. 3. Find the length of BC. 7.5 ANSWER Lesson Quiz

19. ANSWER 37 ft Lesson Quiz 4. A tree casts a shadow that is 30 feet long. At the same time a person standing nearby, who is fivefeet twoinches tall, casts a shadow that is 50 inches long. How tall is the tree to the nearest foot?