Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
6.4 Prove Triangles Similar by AA p Use the AA Similarity Postulate. Goal Your Notes POSTULATE 22: ANGLE-ANGLE (AA) SIMILARITY POSTULATE K L If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Z Y J X nJKL , nXYZ Use the AA Similarity Postulate Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. E D 228 A F Solution B Because they are both right angles, ∠ are congruent. 688 C and ∠ By the Triangle Sum Theorem, 1 1 m∠A 5 1808, so m∠A 5 Therefore, ∠A and ∠ are congruent. . So, nABC , nDEF by the . Checkpoint Determine whether the triangles are similar. If they are, write a similarity statement. 1. P R 2. R 718 788 G 538 318 278 P Q G H 1108 278 788 Q Copyright © Holt McDougal. All rights reserved. F F H Lesson 6.4 • Geometry Notetaking Guide 159 6.4 Prove Triangles Similar by AA p Use the AA Similarity Postulate. Goal Your Notes POSTULATE 22: ANGLE-ANGLE (AA) SIMILARITY POSTULATE K L If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Z Y J X nJKL , nXYZ Use the AA Similarity Postulate Example 1 Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. E 228 A D F 688 Solution C B Because they are both right angles, ∠ B and ∠ E are congruent. By the Triangle Sum Theorem, 688 1 908 1 m∠A 5 1808, so m∠A 5 228 . Therefore, ∠A and ∠ D are congruent. So, nABC , nDEF by the AA Similarity Postulate . Checkpoint Determine whether the triangles are similar. If they are, write a similarity statement. 1. P R 2. R 718 788 G 538 318 278 P Q G H 1108 278 788 Q F nFGH , nRQP Copyright © Holt McDougal. All rights reserved. F H not similar Lesson 6.4 • Geometry Notetaking Guide 159 Your Notes Show that triangles are similar Example 2 Show that the two triangles are similar. a. nRTV and nRQS b. nLMN and nNOP M R O T 498 V 498 Q L S N P Solution a. You may find it helpful to redraw the triangles separately. and m∠ both equal 498, . By the Reflexive Property, Because m∠ ∠ >∠ ∠R > ∠ . So, nRTV , nRQS by the b. The diagram shows ∠L > ∠ } MN i so ∠ >∠ Angles Postulate. . . It also shows that by the Corresponding So, nLMN , nNOP by the . Checkpoint Complete the following exercise. 3. Show that nBCD , nEFD. F C D B 160 Lesson 6.4 • Geometry Notetaking Guide E Copyright © Holt McDougal. All rights reserved. Your Notes Show that triangles are similar Example 2 Show that the two triangles are similar. a. nRTV and nRQS b. nLMN and nNOP M R O T 498 V 498 Q L S N P Solution a. You may find it helpful to redraw the triangles separately. Because m∠ RTV and m∠ Q both equal 498, ∠ RTV > ∠ Q . By the Reflexive Property, ∠R > ∠ R . So, nRTV , nRQS by the AA Similarity Postulate . b. The diagram shows ∠L > ∠ ONP . It also shows that } } MN i OP so ∠ LNM > ∠ P by the Corresponding Angles Postulate. So, nLMN , nNOP by the AA Similarity Postulate . Checkpoint Complete the following exercise. 3. Show that nBCD , nEFD. F C D B E Because they are both right angles, ∠C > ∠F. You know that ∠CDB > ∠FDE by the Vertical Angles Congruence Theorem. So, nBCD , nEFD by the AA Similarity Postulate. 160 Lesson 6.4 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Example 3 Using similar triangles Height A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet long. How tall is the chair? Solution The lifeguard and the chair form sides of two right triangles with the ground, as shown below. The sun’s rays hit the lifeguard and the chair at the same angle. You , so the triangles have two pairs of congruent are similar by the . 6 ft 4 in. x ft 48 in. 6 ft You can use a proportion to find the height x. Write 6 feet 4 inches as inches so you can form two ratios of feet to inches. x ft 5 Write proportion of side lengths. x5 Cross Products Property x5 Solve for x. The chair is feet tall. Checkpoint Complete the following exercise. 4. In Example 3, how long is the shadow of a person that is 4 feet 9 inches tall? Homework Copyright © Holt McDougal. All rights reserved. Lesson 6.4 • Geometry Notetaking Guide 161 Your Notes Example 3 Using similar triangles Height A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet long. How tall is the chair? Solution The lifeguard and the chair form sides of two right triangles with the ground, as shown below. The sun’s rays hit the lifeguard and the chair at the same angle. You have two pairs of congruent angles , so the triangles are similar by the AA Similarity Postulate . 6 ft 4 in. x ft 48 in. 6 ft You can use a proportion to find the height x. Write 6 feet 4 inches as 76 inches so you can form two ratios of feet to inches. x ft 5 76 in. 6 ft Write proportion of side lengths. 48 in. 48 x 5 456 x 5 9.5 Cross Products Property Solve for x. The chair is 9.5 feet tall. Checkpoint Complete the following exercise. 4. In Example 3, how long is the shadow of a person that is 4 feet 9 inches tall? Homework 3 feet Copyright © Holt McDougal. All rights reserved. Lesson 6.4 • Geometry Notetaking Guide 161