Question

consider the diagram and proof by contradiction. given: △abc with ∠b ≅ ∠c prove: ( overline{ab} cong overline{ac} ) which would prove that ( overline{ab} cong overline{ac} )? ( \bigcirc ) converse of the isosceles triangle theorem ( \bigcirc ) substitution ( \bigcirc ) definition of congruency ( \bigcirc ) converse of the triangle parts relationship theorem

Explanation:

The isosceles triangle theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. The converse of this theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. In $\triangle ABC$, we are given that $\angle B \cong \angle C$. The side opposite $\angle B$ is $AC$, and the side opposite $\angle C$ is $AB$. By the converse of the isosceles triangle theorem, we can conclude that $AB \cong AC$. Substitution is a method of replacing a variable or expression with another, which is not relevant here. The definition of congruency refers to the condition for two figures to be congruent (same shape and size), not a theorem to prove side congruence from angle congruence. The converse of the triangle parts relationship theorem is not a standard theorem related to this specific proof; the key here is the converse of the isosceles triangle theorem.

Answer:

A. converse of the isosceles triangle theorem