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Using SAS Similarity Theorem to Determine Triangle Similarity

Learn how to use the SAS Similarity Theorem to determine if triangles are similar by comparing angle measures and side lengths.

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Using SAS Similarity Theorem to Determine Triangle Similarity

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  1. EXAMPLE 3 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?

  2. Both m A andm F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F. The lengths of the sides that include Aand F are proportional. 15 3 AC 3 AB 9 = = 10 2 FH 2 FG 6 ~ = = EXAMPLE 3 Use the SAS Similarity Theorem SOLUTION Shorter sides Longer sides

  3. ANSWER So, by the SAS Similarity Theorem, ABC~FGH. Yes, you can make the right end similar to the left end of the shelter. EXAMPLE 3 Use the SAS Similarity Theorem

  4. 18 9 3 CA 3 BC 5 CD 30 5 15 EC = = = = The corresponding side lengths are proportional. The included angles ACB and DCEare congruent because they are vertical angles. So, ACB ~DCE by the SAS Similarity Theorem. EXAMPLE 4 Choose a method Tell what method you would use to show that the triangles are similar. SOLUTION Find the ratios of the lengths of the corresponding sides. Shorter sides Longer sides

  5. 3. SRT ~ PNQ RT 4 28 3 NQ 21 SR 24 4 = = = = PN 18 3 for Examples 3 and 4 GUIDED PRACTICE Explain how to show that the indicated triangles are similar. SOLUTION Find the ratios of the lengths of the corresponding sides. Shortest sides Longer sides

  6. ANSWER The corresponding side lengths are proportional. The included angles R and Nare right angles. So, SRI ~PNQ by the SAS Similarity Theorem. for Examples 3 and 4 GUIDED PRACTICE Explain how to show that the indicated triangles are similar.

  7. 4. XZW ~ YZX 4 XZ 16 WZ 4 3 YZ XZ 12 3 = = 12 = = 9 for Examples 3 and 4 GUIDED PRACTICE Explain how to show that the indicated triangles are similar. SOLUTION Find the ratios of the lengths of the corresponding sides.

  8. 20 = = 15 WXY XZY = 90° 4 WX XY 3 ANSWER The corresponding side lengths are proportional. The angles WZX and XZYare right angles. So, XZW ~YZX by the SAS and SSS Similarity Theorem. for Examples 3 and 4 GUIDED PRACTICE Explain how to show that the indicated triangles are similar.

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