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Bell Ringer Proving Triangles are Similar by AA,SS, & SAS Example 1 Use the AA Similarity Postulate Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. SOLUTION If two pairs of angles are congruent, then the triangles are similar. 1. G L because they are both marked as right angles. mF + 90° + 61° = 180° mF + 151° = 180° mF = 29° Triangle Sum Theorem Add. Subtract 151° from each side. Example 2 Use the AA Similarity Postulate Are you given enough information to show that RST is similar to RUV? Explain your reasoning. SOLUTION Redraw the diagram as two triangles: RUV and RST. From the diagram, you know that both RST and RUV measure 48°, so RST RUV. Also, R R by the Reflexive Property of Congruence. By the AA Similarity Postulate, RST ~ RUV. Now You Try Use the AA Similarity Postulate Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. ANSWER yes; RST ~ MNL ANSWER yes; GLH ~ GKJ 2. Example 3 Use Similar Triangles A hockey player passes the puck to a teammate by bouncing the puck off the wall of the rink, as shown below. According to the laws of physics, the angles that the path of the puck makes with the wall are congruent. How far from the wall will the teammate pick up the pass? DE AB x 25 = = EC BC 28 40 x · 40 = 25 · 28 Write a proportion. Substitute x for DE, 25 for AB, 28 for EC, and 40 for BC. Cross product property Example 3 Use Similar Triangles 40x = 700 Multiply. 40x 700 = 40 40 Divide each side by 40. x = 17.5 ANSWER Simplify. The teammate will pick up the pass 17.5 feet from the wall. Now Checkpoint You Try Use Similar Triangles Write a similarity statement for the triangles. Then find the value of the variable. 3. ANSWER ABC ~ DEF; 9 4. ANSWER ABD ~ EBC; 3 Example 1 Use the SSS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A. SOLUTION Find the ratios of the corresponding sides. SU 6 6÷6 1 = PR 12 = 12 ÷ 6 = 2 UT 5 5÷5 1 = = = RQ 10 ÷ 5 2 10 TS 4 4÷4 1 = = = QP 8÷4 2 8 All three ratios are equal. So, the corresponding sides of the triangles are proportional. Example 1 Use the SSS Similarity Theorem ANSWER By the SSS Similarity Theorem, PQR ~ STU. The scale factor of Triangle B to Triangle A is . 1 2 Example 2 Use the SSS Similarity Theorem Is either DEF or GHJ similar to ABC? SOLUTION 1. Look at the ratios of corresponding sides in ABC and DEF. Shortest sides Longest sides DE 4 2 = = AB 6 3 FD 8 2 = CA 12 = 3 ANSWER Remaining sides Because all of the ratios are equal, ABC ~ DEF. EF 6 2 = = BC 9 3 Example 2 Use the SSS Similarity Theorem 2. Look at the ratios of corresponding sides in ABC and GHJ. Shortest sides GH AB ANSWER = 6 1 = 1 6 Longest sides JG CA = 14 7 = 6 12 Remaining sides HJ 10 = BC 9 Because the ratios are not equal, ABC and GHJ are not similar. Is either DEF or GHJ similar to ABC? NowCheckpoint You Try Use the SSS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. ANSWER yes; ABC ~ DFE ANSWER no 2. Example 3 Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. SOLUTION C and F both measure 61°, so C F. Compare the ratios of the side lengths that include C and F. DF 5 = Shorter sides AC 3 FE 10 5 = Longer sides = CB 6 3 The lengths of the sides that include C and F are proportional. ANSWER By the SAS Similarity Theorem, ABC ~ DEF. Example 4 Similarity in Overlapping Triangles Show that VYZ ~ VWX. SOLUTION Separate the triangles, VYZ and VWX, and label the side lengths. V V by the Reflexive Property of Congruence. Shorter sides Longer sides VW 4 1 4 = = = VY 4+8 12 3 XV 5 1 5 = = = ZV 5 + 10 15 3 Example 4 Similarity in Overlapping Triangles The lengths of the sides that include V are proportional. ANSWER By the SAS Similarity Theorem, VYZ ~ VWX. NowCheckpoint You Try Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 3. ANSWER No; H M but 12 8 . ≠ 6 8 NowCheckpoint You Try Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 4. ANSWER PQ 3 PR 1 5 1 , Yes; P P, and the = = = = ; PS 2 6 PT 10 2 lengths of the sides that include P are proportional, so PQR ~ PST by the SAS Similarity Theorem. Page 375 #s 8-26 & Page 382 #s 6-18 even only