5.2.2 Inequalities in Two Triangles Key Objectives • Apply inequalities in two triangles. Theorems, Postulates, Corollaries, and Properties • Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. • Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another trian- gle and the third sides are not congruent, then the larger included angle is across from the longer third side. The Hinge Theorem and its converse give information about the relative size of either two angles in a pair of triangles or two sides in a pair of triangles. With these theorems, inequalities involving the lengths of sides or the measures of angles in two triangles can be written to give information about the two triangles. Example 1 Using the Hinge Theorem and Its Converse In these examples, the Hinge Theorem and its converse are used to write inequalities involving the lengths of sides of two triangles or the measures of angles in two triangles. Begin by determining that the Hinge Theorem or the Converse of the Hinge Theorem apply to the two triangles, then write an appropriate inequality.
According to the Hinge Theorem, when two sides of one triangle are congruent to two sides of another triangle, the triangle with the larger included angle will have the longer third side. According to the Converse of the Hinge Theorem, when two sides of one triangle are congruent to two sides of another triangle, the triangle with the longer third side has the larger included angle.
The measures of two angles in two triangles are compared in this example. Begin by comparing the side lengths of the two triangles. That comparison shows that DC = AC (both equal 11 units), BC = BC (because they are the same side) and DB > AB (since 6 > 5). So, △ ABC has two sides congruent to △ BDC . The third side of △ BDC is larger than the third side of △ ABC , so by the Converse of the Hinge Theorem, the measure of the included angle BCD is larger than the measure of the included angle ACB .
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