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Proving Triangles Similar LESSON 7-3 Additional Examples MX AB. Explain why the triangles are similar. Write a similarity statement. Because MX AB, AXM and BXK are both right angles, so AXM BXK. A B because their measures are equal. AMX ~ BKX by the Angle-Angle Similarity Postulate (AA ~ Postulate). Quick Check HELP GEOMETRY Proving Triangles Similar LESSON 7-3 Additional Examples Explain why the triangles must be similar. Write a similarity statement. YVZ WVX because they are vertical angles. VY 12 1 VZ 18 1 = = and = = , so corresponding sides are proportional. VW 24 2 VX 36 2 Therefore, YVZ ~ WVX by the Side-Angle-Side Similarity Theorem (SAS Similarity Theorem). Quick Check HELP GEOMETRY Proving Triangles Similar LESSON 7-3 Additional Examples ABCD is a parallelogram. Find WY. Because ABCD is a parallelogram, AB || DC. XAW ZYW and AXW YZW because parallel lines cut by a transversal form congruent alternate interior angles. Therefore, AWX ~ YWZ by the AA ~ Postulate. Use the properties of similar triangles to find WY. WY WZ = WA WX 10 WY = 4 5 10 WY = 4 5 WY = 12.5 HELP Corresponding sides of ~ triangles are proportional. Substitute. Solve for WY. Quick Check GEOMETRY Proving Triangles Similar LESSON 7-3 Additional Examples Joan places a mirror 24 ft from the base of a tree. When she stands 3 ft from the mirror, she can see the top of the tree reflected in it. If her eyes are 5 ft above the ground, how tall is the tree? Draw the situation described by the example. TR represents the height of the tree, point M represents the mirror, and point J represents Joan’s eyes. Both Joan and the tree are perpendicular to the ground, so m JOM = mTRM, and therefore JOM TRM. The light reflects off a mirror at the same angle at which it hits the mirror, so JMO TMR. Use similar triangles to find the height of the tree. HELP GEOMETRY Proving Triangles Similar LESSON 7-3 Additional Examples (continued) JOM ~ TRM RM TR = OM JO TR = 24 3 5 24 TR = 3 5 5 TR = 40 The tree is 40 ft tall. HELP AA ~ Postulate Corresponding sides of ~ triangles are proportional. Substitute. Solve for TR. Quick Check GEOMETRY