5.2.2 Inequalities in Two Triangles (continued)
The lengths of two sides in two triangles are compared in this example. Begin by comparing the side lengths of the two triangles. That comparison shows that RQ = PS (both equal to 10 units), and QS = SQ (they are the same side). Therefore, △ PQS has two sides congruent to two sides of △ RSQ . The included angle RQS in the one triangle is given to be greater than the included angle RSQ in the other triangle, so by the Hinge Theorem, the opposite side RS is greater than the opposite side PQ . The Hinge Theorem and its converse produce inequalities that can be used to determine the range of values for the measure of an angle. To find the range of values for z , first compare the side lengths of the two triangles. MN = KN , because the sides are congruent. Also, LN = LN because they are the same side. Finally, ML < KL (since 16 < 17). Since ∠ LNM is in a triangle, ∠ LNM must be greater than zero. Therefore, m ∠ LNM > 0. Substitute the expression given in the figure for m ∠ LNM into this inequality. m ∠ LNM > 0 2 z − 6 > 0. Solve the inequality for z . The result is z > 3. By the converse of the Hinge Theorem, ∠ LNM must be less than ∠ KNL and, as given, m ∠ KNL = 90 ° . Therefore, m ∠ LNM < 90 ° . Substitute the expression given in the figure for m ∠ LNM into this inequality. m ∠ LNM < 90 2 z − 6 > 90. Solve the inequality for z . The result is z < 48. Combining the inequalities yields 3 < z < 48 as the answer.
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