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 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 AA Similarity Angle-Angle similarity . When two triangles have corresponding angles that are congruent as shown below, the triangles are similar. 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, congruent. D and G are 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. b. ∆ABE and ∆ACD ∆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 ANSWER So, ∆ ABE ~ ∆ ACD by the AA Similarity Postulate. A. 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. ANSWER So, ∆SVR ~ ∆UVT by the AA Similarity Postulate. GUIDED PRACTICE for Examples 1 and 2 Show that the triangles are similar. Write a similarity statement. 1. ∆FGH and ∆RQS ANSWER In each triangle all three angles measure 60°, so by the AA similarity postulate, the triangles are similar ∆FGH ~ ∆QRS. GUIDED PRACTICE for Examples 1 and 2 Show that the triangles are similar. Write a similarity statement. 2. ∆CDF and ∆DEF ANSWER 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. GUIDED PRACTICE 3. for Examples 1 and 2 Reasoning Suppose in Example 2, part (b), SR triangles still be similar? Explain. TU . Could the ANSWER Yes; if S T, the triangles are similar by the AA Similarity Postulate. 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. 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 = 50 ft 40 in. 64 in. 40x = 64(50) x = 80 Write proportion of side lengths. Cross Products Property Solve for x. ANSWER The flagpole is 80 feet tall. The correct answer is C. 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. for Example 3 GUIDED PRACTICE 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 tree height your height = length of shadow length of your shadow