Question

given: in triangle abc, ∠b ≅ ∠c
prove: \overline{ab} ≅ \overline{ac}

complete the paragraph proof.
we are given that ∠b ≅ ∠c. assume segment ab is not congruent to \boxed{\quad}.
if ab > ac, then m∠c > m∠b by the \boxed{\quad}.
if ab < ac, then m∠c < m∠b by the converse of the triangle parts relationship theorem.
but by the definition of congruent, we know the measure of angle b equals the measure of \boxed{\quad} by the given statement. therefore, we have a contradiction: ab = ac, and \overline{ab} ≅ \overline{ac}.

Explanation:

Step1: Identify the first blank

The proof aims to show $\overline{AB} \cong \overline{AC}$, so the assumption is $\overline{AB}$ not congruent to $\overline{AC}$.

Step2: Identify the second blank

The theorem linking longer side to larger opposite angle is the converse of the triangle parts relationship theorem.

Step3: Identify the third blank

Given $\angle B \cong \angle C$, their measures are equal, so $m\angle B = m\angle C$.

Answer:

1. $\overline{AC}$
2. converse of the triangle parts relationship theorem
3. $\angle C$