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6.5 – Prove Triangles Similar by SSS and SAS Geometry Ms. Rinaldi

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Side-Side-Side (SSS) Similarity Theorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

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SOLUTION EXAMPLE 1 Use the SSS Similarity Theorem Compare ABC and DEF by finding ratios of corresponding side lengths. Shortest sides AB DE 4 3 8 6 == Is either DEF or GHJ similar to ABC ? Longest sides CA FD 4 3 16 12 == Remaining sides BC EF 4 3 12 9 = = All of the ratios are equal, so ABC ~ DEF. ANSWER

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EXAMPLE 1 Use the SSS Similarity Theorem (continued) Longest sides CA JG 16 == 1 Remaining sides BC HJ 6 5 12 10 = = The ratios are not all equal, so ABC and GHJ are not similar. ANSWER Compare ABC and GHJ by finding ratios of corresponding side lengths. Shortest sides AB GH 8 8 == 1

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EXAMPLE 2 Use the SSS Similarity Theorem Which of the three triangles are similar? Write a similarity statement.

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SOLUTION EXAMPLE 3 Use the SSS Similarity Theorem ALGEBRA Find the value of x that makes ABC ~ DEF. STEP 1Find the value of x that makes corresponding side lengths proportional. 4 12 = x –1 18 Write proportion. 4 18 = 12(x – 1) 72 = 12x – 12 7 = x Cross Products Property Simplify. Solve for x.

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EXAMPLE 3 Use the SSS Similarity Theorem (continued) Check that the side lengths are proportional when x = 7. STEP 2 BC = x – 1 = 6 6 18 4 12 = AB DE BC EF = ? DF = 3(x + 1) = 24 8 24 4 12 = AB DE AC DF = ? When x = 7, the triangles are similar by the SSS Similarity Theorem. ANSWER

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EXAMPLE 4 Use the SSS Similarity Theorem Find the value of x that makes XZ Y P R Q 20 12 x + 6 30 3(x – 2) 21

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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.

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EXAMPLE 5 Use the SAS Similarity Theorem Lean-to Shelter You are building a lean-to shelter starting from a tree branch, as shown. Can you construct the right end so it is similar to the left end using the angle measure and lengths shown?

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EXAMPLE 5 Use the SAS Similarity Theorem (continued) Both m A and m F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F. ~ SOLUTION Shorter sidesLonger sides AB FG 3 2 9 6 == AC FH 3 2 15 10 == The lengths of the sides that include A and F are proportional. ANSWER So, by the SAS Similarity Theorem, ABC ~ FGH. Yes, you can make the right end similar to the left end of the shelter.

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EXAMPLE 6 Choose a method Tell what method you would use to show that the triangles are similar. Find the ratios of the lengths of the corresponding sides. Shorter sidesLonger sides SOLUTION CA CD 3 5 18 30 == BC EC 3 5 9 15 == The corresponding side lengths are proportional. The included angles ACB and DCE are congruent because they are vertical angles. So, ACB ~ DCE by the SAS Similarity Theorem.

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EXAMPLE 7 Choose a method A. SRT ~ PNQ Explain how to show that the indicated triangles are similar. B. XZW ~ YZX Explain how to show that the indicated triangles are similar.

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