Question

consider the diagram and proof by contradiction.

given: \\(\triangle abc\\) with \\(\angle b \cong \angle c\\)
prove: \\(\overline{ab} \cong \overline{ac}\\)

which would prove that \\(\overline{ab} \cong \overline{ac}\\)?

• converse of the isosceles triangle theorem
• substitution
• definition of congruency
• converse of the triangle parts relationship theorem

Explanation:

Identify the given conditions and the target statement

Given: \(\triangle ABC\) with \(\angle B \cong \angle C\)
Prove: \(\overline{AB} \cong \overline{AC}\)

Analyze the geometric theorems relating angles and sides

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.

Match the converse theorem to the target proof

Since we are given \(\angle B \cong \angle C\) and want to prove \(\overline{AB} \cong \overline{AC}\), we are using the converse of the isosceles triangle theorem.

Answer:

• converse of the isosceles triangle theorem (Correct answer)
• substitution
• definition of congruency
• converse of the triangle parts relationship theorem