What is one proof of the converse of the Isosceles Triangle Theorem?

1 Answer

See explanation.

Explanation:

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:

  • #hat A cong hat B# (hypotesis of the theorem)
  • #hat(ACP) cong hat(BCP)# since #CP# lies on the bisector of #hat C#
  • #CP# is a shared side between the two triangles

Since the triangles #ACP# and #BCP# are congruent, we conclude that #BC cong AC#.