Chapter 8 Similarity

USING SIMILARITY THEOREMS USING SIMILAR TRIANGLESIN REAL LIFE Chapter 8Similarity Section 8.5 Proving Triangles are Similar

USING SIMILARITY THEOREMS Postulate E D C F B A A D and C F  ABC ~ DEF

AA Similarity Postulate W  W WVX  WZY AA Similarity

USING SIMILARITY THEOREMS AA Similarity Postulate WSU ~ VTU AA Similarity Postulate CAB ~ TQR

USING SIMILARITY THEOREMS THEOREMS P A AB PQ BC QR CA RP Q R If = = B C THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. then ABC ~ PQR.

Using the SSS Similarity Theorem CA FD AB DE BC EF 12 14 A E G J C 6 4 6 9 6 10 F D 8 B H 12 8 = = , 6 4 9 6 3 2 3 2 3 2 = = , = = Shortest sides Longest sides Remaining sides Which of the following three triangles are similar? SOLUTION To decide which of the triangles are similar, consider the ratios of the lengths of corresponding sides. Ratios of Side Lengths ofABC andDEF Because all of the ratios are equal, ABC~ DEF

Using the SSS Similarity Theorem BC HJ CA JG AB GH 12 14 A E E G J C 6 6 4 4 6 9 6 10 F F D D 8 8 B H = = 1, = 6 6 9 10 6 7 12 14 = = , Shortest sides Longest sides Remaining sides Which of the following three triangles are similar? 12 14 A G J C 6 9 6 10 B H SOLUTION To decide which of the triangles are similar, consider the ratios of the lengths of corresponding sides. Ratios of Side Lengths ofABC andGHJ Since ABC is similar to DEF and ABC is not similar to GHJ, DEF is not similar to GHJ. Because all of the ratios are not equal,ABCandDEF are not similar.

USING SIMILARITY THEOREMS THEOREMS X M P N Z Y XY MN ZX PM If XM and= THEOREM 8.3 Side-Angle-Side (SAS) Similarity Theorem 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. then XYZ ~ MNP.

Using the SAS Similarity Theorem SR SP SP = 4, PR = 12, SQ = 5, QT = 15 GIVEN RST ~ PSQ PROVE S 4 5 P Q 12 15 ST SQ SQ + QT SQ 5 + 15 5 20 5 = = = = 4 SP + PR SP 4 + 12 4 16 4 = = = = 4 R T Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that RST ~ PSQ. Use the given lengths to prove thatRST ~ PSQ. SOLUTION Paragraph ProofUse the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides. The side lengths SR and ST are proportional to the corresponding side lengths of PSQ.

USING SIMILARITY THEOREMS Nothing is known about any corresponding congruent angles SSS ~ Theorem is the only choice 6 9 SSS ~ Theorem ABC ~ XYZ

USING SIMILARITY THEOREMS Nothing is known about any corresponding congruent angles SSS ~ Theorem is the only choice Only one Angle is Known Use SAS ~ Theorem 6 9 6 3 SSS ~ Theorem ABC ~ XYZ

USING SIMILARITY THEOREMS Parallel lines give congruent angles Use AA ~ Postulate Only one Angle is Known Use SAS ~ Theorem

USING SIMILARITY THEOREMS No, Need to know the included angle.

USING SIMILARITY THEOREMS 40 No, Need to know the included angle. Yes, AA ~ Postulate DRM ~ XST

USING SIMILARITY THEOREMS SSS ~ Theorem AA ~ Theorem SAS ~ Theorem

HW Pg :6;9;11;13-17;19-25;27-29;32-34;39-47

Chapter 8 Similarity