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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