4.2.2 Triangle Congruence: ASA, AAS, and HL (continued) Example 3 Using AAS to Prove Triangles Congruent
The AAS Congruence Theorem uses two angles and a nonincluded side to show two triangles congruent. For any two angles in a triangle there are two nonincluded sides. Either nonincluded side can be used with the AAS Postulate. However, the nonincluded sides in each triangle must be corresponding sides. It is given that ∠ Q is congruent to ∠ S in this example. It is also given that ∠ QRP and ∠ SRP are right angles. So, they are congruent by the Right Angle Congruence Theorem. Since two angles of one triangle are congruent to two angles of another triangle, ASA or AAS might be used to prove these two triangle congruent. Notice that the two triangles share a common side. This side is the nonincluded side to the congruent angles. Therefore, AAS can be used to prove these two triangles congruent. The HL Congruence Theorem can be used to show that two right triangles are congruent. Remember, the hypotenuse of a right triangle is the side that is opposite of the right angle. The two other sides are called the legs. In this example it is given that the two triangles are right triangles and that their hypotenuses are congruent. In order to prove two right triangles congruent by the HL Theorem, the hypotenuse and a leg of one right triangle must be congruent to the hypotenuse and a leg of the other right triangle. However, in this example the congruence of either pair of legs cannot be deduced. Therefore, the HL Theorem cannot be used to prove that these triangles are congruent.
Example 4 Applying HL Congruence
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