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Proving Triangles are Similar Using Similarity Theorems • In this lesson, you will study 2 alternate ways of proving that two triangles are similar: Side-SideSide Similarity Theorem (SSS) and the Side-Angle-Side Similarity Theorem (SAS). Side Side Side(SSS) Similarity Theorem • If the corresponding sides of two triangles are proportional, then the triangles are similar. P A Q B C AB PQ = BC QR = CA RP R THEN ∆ABC ~ ∆PQR Side Angle Side (SAS)Similarity Thm. • If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. M X P N If X M and Z Y THEN ∆XYZ ~ ∆MNP ZX PM = XY MN Ex. 1: Proof of SSS Theorem •Given: RS LM = •Prove ST MN = TR ∆RST ~ ∆LMN NL Locate P on RS so that PS = LM. Draw PQ so that PQ ║ RT. Then ∆RST ~ ∆PSQ, by the AA Similarity Postulate, and RS LM = ST MN = TR NL Because PS = LM, you can substitute in the given proportion and find that SQ = MN and QP = NL. By the SSS Congruence Theorem, it follows that ∆PSQ ∆LMN Finally, use the definition of congruent triangles and the AA Similarity Postulate to conclude that ∆RST ~ ∆LMN. Ex. 2: Using the SSS Similarity Thm. • Which of the three triangles are similar? E 12 A C 9 6 6 F 4 D 8 B 14 G J 10 6 H To decide which, if any, of the triangles are similar, you need to consider the ratios of the lengths of corresponding sides. Ratios of Side Lengths of ∆ABC and ∆DEF. AB DE = 6 4 = 3 CA 2 FD = 12 8 = 3 BC 2 EF = 9 6 = 3 2 Because all of the ratios are equal, ∆ABC ~ ∆DEF. Ratios of Side Lengths of ∆ABC ~ ∆GHJ AB GH = 6 6 =1 CA JG = 12 14 = 6 BC 7 HJ = 9 10 Because the ratios are not equal, ∆ABC and ∆GHJ are not similar. Since ∆ABC is similar to ∆DEF and ∆ABC is not similar to ∆GHJ, ∆DEF is not similar to ∆GHJ. Ex. 3: Using the SAS Similarity Thm. • Use the given lengths to prove that ∆RST ~ ∆PSQ. S 4 P 12 Given: SP=4, PR = 12, SQ = 5, and QT = 15; 5 Q Prove: ∆RST ~ ∆PSQ Use the SAS Similarity 15 R SR SP + PR 4 + 12 16 = = = SP SP 4 4 T Theorem. Begin by finding the ratios of the lengths of the corresponding sides. = 4 ST SQ + QT 5 + 15 20 = = = SQ SQ 5 5 = 4 So, the side lengths SR and ST are proportional to the corresponding side lengths of ∆PSQ. Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that ∆RST ~ ∆PSQ.