1.2. Two facts from geometry 7 1.2.2. Proposition. If ΔABC and ΔXY Z are similar triangles with ∠A = ∠X, ∠B = ∠Y , and ∠C = ∠Z, then the ratios of corresponding sides are equal. That is, |AB| |AC| = |XY | |XZ| , |AB| |BC| = |XY | |Y Z| , |BC| |AC| = |Y Z| |XZ| . Exercises 1. Find all the angles in Figure 1.2.1 that are equal to ∠BPE. 2. Suppose that the lines AB and CD are parallel as in Figure 1.2.2, and that the lines OQ and OS are drawn. Show that ΔOQS and ΔOPR are similar. Figure 1.2.2 Figure 1.2.3 3. Suppose that in Figure 1.2.3 |AB| = 4, |BC| = 8, and |XZ| = √ 3. Find the lengths of the remaining sides in the two triangles. 4. In Figure 1.2.3, suppose that |AB| = 6, |AC| = 8, |BC| = 10, and |XY | = 3. Find |XZ| and |Y Z|.
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