7.1.3 Triangle Similarity: AA, SSS, and SAS (continued) Example 4 Writing Proofs with Similar Triangles
Two triangles are proved to be similar in this example. Since Q is the midpoint of PR and T is the midpoint of PS as given, then PQ = QR and PT = TS . By the Segment Addition Postulate, PR = PQ + QR and PS = PT + TS . Substituting gives PR = PQ + PQ and PS = PT + PT . These equations are rearranged to find the ratios of two corresponding sides in the two triangles. Both these ratios are equal to 2. The angles between the proportional sides of the two triangles are congruent, because they are the same angle, P . Therefore, the triangles are similar by Side-Angle-Side Similarity. Two triangles are proved to be similar and the length of an unknown side is derived from the similarity ratio. It is given that LC || SE . The lengths of IS , LC , and SE are given. By the Corresponding Angles Theorem, ∠ ILC ≅ ∠ ISE (or ∠ ICL ≅ ∠ IES ). Because they are the same angle, ∠ I ≅ ∠ I . Therefore, by Angle- Angle Similarity, △ SEI ∼ △ LCI . Find the unknown side length, LI , by applying the similarity ratio for the triangles. Set up a proportion using the known lengths SI and SE , and the corresponding lengths LI and LC . Substitute the known lengths of sides, cross multiply, and solve the resulting equation for the length of LI . The solution yields LI = 4 cm.
Example 5 Problem-Solving Application
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