6.4 Prove Triangles Similar by AA

1. 6.4 Prove Triangles Similar by AA Before: You used the AAS Congruence Theorem Now: You will use the AA Similarity Postulate Why? So you can use similar triangles to understand aerial photography

3. Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. EXAMPLE 1 Use the AA Similarity Postulate

4. 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. ANSWER So, ∆CDE~∆KGHby the AA Similarity Postulate. EXAMPLE 1 Use the AA Similarity Postulate SOLUTION

5. b. a. ∆ABEand ∆ACD ∆SVRand ∆UVT EXAMPLE 2 Show that triangles are similar Show that the two triangles are similar.

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

7. b. You know SVRUVTby the Vertical Angles Congruence Theorem. The diagram shows RS||UTso SUby the Alternate Interior Angles Theorem. ANSWER So, ∆SVR~ ∆UVTby the AA Similarity Postulate. EXAMPLE 2 Show that triangles are similar SOLUTION

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. for Examples 1 and 2 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. for Examples 1 and 2 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. for Examples 1 and 2 GUIDED PRACTICE

11. EXAMPLE 3 Standardized Test Practice

12. EXAMPLE 3 Standardized Test Practice 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 Standardized Test Practice 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 for Example 3 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 for Example 3 GUIDED PRACTICE