4.2.2 Triangle Congruence: ASA, AAS, and HL Key Objectives • Apply ASA, AAS, and HL to construct triangles and to solve problems. • Prove triangles congruent by using ASA, AAS, and HL. Key Terms • An included side is the common side of two consecutive angles in a polygon. Theorems, Postulates, Corollaries, and Properties
• Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. • Angle-Angle-Side (AAS) Congruence Theorem If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. • Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Example 1 Problem Solving with ASA
The common side of two consecutive angles in a polygon is the included side. The ASA Congruence Postulate allows for two triangles to be proven congruent when two angles and the included side in one triangle are congruent to two angles and the included side in the other triangle.
Example 2 Applying ASA Congruence
The ASA Congruence Postulate can be used to prove two triangles congruent when two angles and their included side in one triangle are congruent to two angles and their included side in the other triangle. In this example it is given that the triangles have one pair of congruent sides. So, to prove that these triangles are congruent by ASA, show that the angles on each end of the congruent sides are congruent.
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