7.1.3 Triangle Similarity: AA, SSS, and SAS (continued)
Two triangles are determined to be similar in this example. The lengths of the sides of the triangles are given. Calculate the ratios of the pairs of corresponding sides of the two triangles. Each of the ratios of the sides is 3/2, so the triangles are similar, according to the Side-Side-Side Similarity Theorem. If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. Two triangles are determined to be similar in this example. The lengths of two corresponding sides of the triangles are given. The triangles share a common angle. By the Reflexive Property of Congruence, ∠ D ≅ ∠ D . Calculate the ratios of the pairs of corresponding sides of the two triangles. Each of the two ratios of the sides is 6. ∠ D is the angle included between the two proportional sides, so the triangles are similar, according to the Side-Angle- Side Similarity Theorem.
Example 3 Finding Lengths in Similar Triangles
The length of an unknown side in a triangle is found in this example using the similarity ratio for two triangles. It is given that PQ || ST , and the lengths PR , QR , and RT are given. According to the Alternate Interior Angles Theorem, ∠ Q ≅ ∠ S (because PQ || ST ). By the Vertical Angles Theorem, ∠ PRQ ≅ ∠ TRS . Therefore, △ PQR ∼ △ TSR by Angle-Angle Similarity. Now, find the unknown side length SR by applying the similarity ratio for the triangles. Set up a proportion using the known corresponding lengths RT and PR and the corresponding lengths SR and QR . Substitute the known lengths of sides, cross multiply, and solve the resulting equation for the length of SR . The solution yields SR = 36.
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