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Rule

Converse Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent.

Isosceles triangle with a movable point that can be used to build triangles with different pairs of congruent angles.

Based on this diagram, the following relation holds true.

$∠ B ≅ ∠ C$ $\Rightarrow$ $\overline{A B} ≅ \overline{A C}$

This theorem is the converse theorem to the Isosceles Triangle Theorem. It is also known as the Converse Base Angles Theorem.

Proof

Consider a triangle $A B C$ with two congruent angles.

Let $P$ be the point of intersection of $\overline{B C}$ and the angle bisector of $∠ A .$ Since $\overline{A P}$ is the angle bisector of $∠ A,$ then $∠ B A P ≅ ∠ C A P .$

By the Reflexive Property of Congruence,

\overline{A P}

△ A B P

is congruent to

\overline{A P}

△ A C P .

Because of the Angle-Angle-Side Congruence Theorem, both triangles are congruent.

△ A B P ≅ △ A C P

Since corresponding parts of congruent triangles are congruent, it follows that

A B

is congruent to

A C .

\overline{A B} ≅ \overline{A C}

It has been proven that if two angles of a triangle are congruent, then the sides opposite them are congruent.

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Proof