4.2.1 Triangle Congruence: SSS and SAS Key Objectives • Apply SSS and SAS to construct triangles and to solve problems. • Prove triangles congruent by using SSS and SAS. Key Terms • The property of triangle rigidity states that if the side lengths of a triangle are given, the triangle can have only one shape. • An included angle is an angle formed by two adjacent sides of a polygon. Theorems, Postulates, Corollaries, and Properties • Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. • Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Example 1 Using SSS to Prove Triangle Congruence

By the property of triangle rigidity, if the sides of a triangle are known, then that triangle can have only one shape. It follows that if the sides of another triangle are also known, and those sides are all congruent to the first triangle’s sides, then the two triangles must have the same shape. Any two figures with the same size and shape are congruent. Therefore, if the sides of one triangle are congruent to the sides of another triangle, then those two triangles must be congruent. To show that two triangles are congruent by SSS, just show that the three pairs of corresponding sides are congruent. In this example, the given figure is a quadrilateral (four-sided polygon) with a segment NP that divides it into two triangles. In the figure it is given that each pair of opposite sides of the quadrilateral are congruent, MN ≅ OP and MP ≅ ON . So, two sides of △ MNP are congruent to two sides of △ OPN . The third side of each triangle is formed by the diagonal segment, NP . NP is a side for each triangle and it is congruent to itself by the Reflexive Property of Congruence. So, since the three sides of △ MNP are congruent to the corresponding three sides of △ OPN , △ MNP ≅ △ OPN by SSS.

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