question archive The converse of the Isosceles Triangle Theorem states that if two angles ##hat A## and ##hat B## of a triangle ##ABC## are congruent, then the two sides ##BC## and ##AC## opposite to these angles are congruent
The converse of the Isosceles Triangle Theorem states that if two angles ##hat A## and ##hat B## of a triangle ##ABC## are congruent, then the two sides ##BC## and ##AC## opposite to these angles are congruent
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The converse of the Isosceles Triangle Theorem states that if two angles ##hat A## and ##hat B## of a triangle ##ABC## are congruent, then the two sides ##BC## and ##AC## opposite to these angles are congruent.
The proof is very quick: if we trace the bisector of ##hat C## that meets the opposite side ##AB## in a point ##P##, we get that the angles ##hat(ACP)## and ##hat(BCP)## are congruent.
We can prove that the triangles ##ACP## and ##BCP## are congruent. In fact, the hypotheses of the AAS criterion are satisfied:
1)##hat A cong hat B## (hypotesis of the theorem)
2)##hat(ACP) cong hat(BCP)## since ##CP## lies on the bisector of ##hat C##
3)##CP## is a shared side between the two triangles
Since the triangles ##ACP## and ##BCP## are congruent, we conclude that ##BC cong AC##.
