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Transcript

Proving Triangle Congruence By Angle-Side-Angle and AngleAngle-Side We already know that you can prove triangle congruence with two methods, Side-Side-Side (SSS) and Side-Angle-Side (SAS). SSS requires three sides. SAS requires two sides and the included angle. However, there are two other ways to prove triangle congruence. One way uses two angles and their included side. This method is called Angle-Side-Angle (ASA). Another method uses two angles and a nonincluded side. This is called Angle-Angle-Side (AAS). ASA is a postulate, however AAS is a theorem. They are both written below. Proving the Angle-Angle-Side Theorem To prove the AAS Theorem, we have to use one of the previous postulates we’ve learned in this chapter. This leaves us with SSS, SAS, and ASA. Using we can’t use SSS because we only have one side. The same goes for using SAS. That leaves ASA. To use ASA we will have to first prove that C Z. To do this we can use Theorem 4-1. This theorem states that if two angles of one triangle are congruent to two angles of another triangle, the third angles of both triangles are congruent. In this case, the two congruent pairs of angles are angles A and X and angles B and Y. Using this theorem, we can prove that C Z. Now that we know this, we can prove the two triangles congruent using ASA. The textbook’s proof is below. They chose to make a flow proof. Extra Problems