1. In ABC and XZW, m A = m X and m B = m Z 1. In ABC and XZW, m A = m X and m B = m Z. What can you conclude about m C and m W? ANSWER They are the same. 2. Solve = . x 18 54 9 ANSWER 108

3. ABC DEF. Find x. ~ ANSWER 10

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

EXAMPLE 1 Use the AA Similarity Postulate SOLUTION Because they are both right angles, D and G are congruent. By the Triangle Sum Theorem, 26° + 90° + m E = 180°, so m E = 64°. Therefore, E and H are congruent. ANSWER So, ∆CDE ~ ∆KGH by the AA Similarity Postulate.

EXAMPLE 2 Show that triangles are similar Show that the two triangles are similar. a. ∆ABE and ∆ACD b. ∆SVR and ∆UVT

EXAMPLE 2 Show that triangles are similar SOLUTION a. You may find it helpful to redraw the triangles separately. Because m ABE and m C both equal 52°, ABE C. By the Reflexive Property, A A. So, ∆ ABE ~ ∆ ACD by the AA Similarity Postulate. ANSWER

EXAMPLE 2 Show that triangles are similar SOLUTION b. You know SVR UVT by the Vertical Angles Congruence Theorem. The diagram shows RS ||UT so S U by the Alternate Interior Angles Theorem. So, ∆SVR ~ ∆UVT by the AA Similarity Postulate. ANSWER

GUIDED PRACTICE for Examples 1 and 2 Show that the triangles are similar. Write a similarity statement. 1. ∆FGH and ∆RQS In each triangle all three angles measure 60°, so by the AA similarity postulate, the triangles are similar ∆FGH ~ ∆QRS. ANSWER

GUIDED PRACTICE for Examples 1 and 2 Show that the triangles are similar. Write a similarity statement. 2. ∆CDF and ∆DEF Since m CDF = 58° by the Triangle Sum Theorem and m DFE = 90° by the Linear Pair Postulate the two triangles are similar by the AA Similarity Postulate; ∆CDF ~ ∆DEF. ANSWER

GUIDED PRACTICE for Examples 1 and 2 3. Reasoning Suppose in Example 2, part (b), SR TU . Could the triangles still be similar? Explain. Yes; if S T, the triangles are similar by the AA Similarity Postulate. ANSWER

EXAMPLE 3 Standardized Test Practice

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.

Standardized Test Practice 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. x ft 64 in. 50 ft 40 in. = Write proportion of side lengths. 40x = 64(50) Cross Products Property x = 80 Solve for x. The flagpole is 80 feet tall. The correct answer is C. ANSWER

GUIDED PRACTICE for Example 3 4. What If ? A child who is 58 inches tall is standing next to the woman in Example 3. How long is the child’s shadow? ANSWER 36.25 in.

GUIDED PRACTICE for Example 3 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. tree height your height = length of your shadow length of shadow SAMPLE ANSWER

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

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

Daily Homework Quiz 3. Find the length of BC ANSWER 7.5

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