4.1.3 Congruent Triangles Key Objectives • Use properties of congruent triangles. • Prove triangles congruent by using the definition of congruence. Key Terms • Corresponding angles of polygons are the angles in the same relative position in two different polygons that have the same number of angles. • Corresponding sides of polygons are the sides in the same relative position in two different polygons that have the same number of sides. • Two polygons are congruent polygons if and only if their corresponding angles and sides are congruent. Theorems, Postulates, Corollaries, and Properties • Corresponding Angles of Congruent Polygons Property The corresponding angles of congruent polygons are congruent. • Corresponding Sides of Congruent Polygons Property The corresponding sides of congruent polygons are congruent. Example 1 Naming Congruent Corresponding Parts In a congruence statement, such as △ ABC ≅ △ DEF , each polygon’s vertices must be listed in the order of the corresponding parts. So, a congruence statement indicates not only that the two polygons are congruent, but also which parts of those two polygons are corresponding.

In this example, it is given that the two triangles are congruent. Therefore, by the properties of congruent polygons, the corresponding parts (sides and angles) of △ ABC and △ DEF must also be congruent. The corresponding parts are the sides or angles that are in the same relative position. Note that the corresponding parts of the two triangles could be identified using only the triangle’s names given in the congruence statement.

Example 2 Using Corresponding Parts of Congruent Triangles

In this example, it is given that △ ABD is congruent to △ CBD . It is also given in the figure that ∠ CBD is a right angle. So, m ∠ CBD = 90°. It follows that m ∠ ABD = 90° by the Supplementary Angles Theorem. Alternatively, it can be deduced that m ∠ ABD = 90° because ∠ ABD and ∠ CBD are corresponding angles in congruent triangles and m ∠ CBD = 90°. The expression given in the figure for m ∠ ABD is (8 x + 2)°. Substitute this expression for m ∠ ABD into the equation m ∠ ABD = 90° and solve for x .

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