4.2.1 Triangle Congruence: SSS and SAS (continued) Example 2 Using SAS to Prove Triangle Congruence
By the SAS Congruence Postulate, two triangles are congruent when two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle. Note that the included angle is the angle formed by two adjacent sides. To show that two triangles are congruent by SAS, show that two sides and their included angle in the first triangle are congruent to two sides and their included angle in the second triangle. In this example, one pair of angles and one pair of sides are given to be congruent in △ ACB and △ ACD . So, to use SAS to show that these two triangles are congruent, show that a second pair of sides are congruent. However, the pair of sides must be the sides that make △ BAC and △ DAC the included angle.
Example 3 Verifying Triangle Congruence
The lengths of the sides of △ YXZ are unknown in this example. But expressions that represent the length of each side of △ YXZ are given. XZ = x + 1 YZ = x + 4 XY = 2 x − 3 Substitute 5 for x into each expression and simplify.
XZ = 5 + 1 = 6 YZ = 5 + 4 = 9 XY = 2(5) − 3 = 7
Therefore, VW = XZ , UW = YZ , and UV = XY . So, by the definition of congruence, each of these pairs of sides is congruent. Since the three sides of △ UVW are congruent to the three sides of △ YXZ , the two triangles are congruent by SSS.
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