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

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

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

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

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

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

Using the SSS Similarity Theorem Which of the following three triangles are similar? E F D 8 6 4 A C B 12 9 G J H 14 10 A 12 C E F D 8 6 4 G 14 J 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. = = 1, 6 AB GH Shortest sides = = , 12 14 CA JG 6 7 Longest sides = 9 10 BC HJ Remaining sides Because all of the ratios are not equal,  ABC and  DEF are not similar.

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

Use the given lengths to prove that  RST ~  PSQ. Using the SAS Similarity Theorem Use the given lengths to prove that  RST ~  PSQ. SOLUTION GIVEN SP = 4, PR = 12, SQ = 5, QT = 15 PROVE  RST ~  PSQ P Q S R T Paragraph Proof Use the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides. 12 4 5 15 = = = = 4 SR SP 16 4 SP + PR 4 + 12 = = = = 4 ST SQ 20 5 SQ + QT 5 + 15 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.

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

Only one Angle is Known Use SAS ~ Theorem 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 9 6 6 3 SSS ~ Theorem ABC ~ XYZ

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

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

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

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