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 problem asks to prove $\overline{AB} \cong \overline{AC}$ given $\angle B \cong \angle C$ in $\triangle ABC$. The converse of the isosceles triangle theorem states that if two angles of a triangle are congruent, then the sides opposite them are congruent. This directly applies here: $\angle B$ and $\angle C$ are opposite $\overline{AC}$ and $\overline{AB}$ respectively, so their congruence implies the sides are congruent. Other options do not fit: substitution is a method, not a theorem for this conclusion; definition of congruency describes equal measure but doesn’t link angles to sides; the converse of the triangle parts relationship theorem relates to inequalities, not congruence.

Answer:

A. converse of the isosceles triangle theorem