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
if ab > ac, then
if ab < ac, then m
the converse of the
triangle parts rela
but by the definiton of congruent, we know the measure of angle b equals the measure of
by the given statement. therefore, we have a contradiction: ab = ac, and \overline{ab} ≅ \overline{ac}.

Explanation:

Step1: Identify the first blank

We are trying to prove \( \overline{AB} \cong \overline{AC} \) by contradiction. So we assume \( AB \) is not congruent to \( AC \). The first blank should be \( \overline{AC} \) (segment \( AC \)) because we are assuming the opposite of what we want to prove, i.e., \( AB
ot\cong AC \).

Step2: Identify the second part (if \( AB > AC \))

If \( AB > AC \), then by the "Converse of the Triangle Angle - Side Relationship" (or more precisely, the theorem that if one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side), we would have \( m\angle C > m\angle B \). But we know \( \angle B \cong \angle C \), so \( m\angle B=m\angle C \), which is a contradiction.

Step3: Identify the third blank (angle part)

By the definition of congruent angles, we know the measure of angle \( B \) equals the measure of angle \( C \) (since \( \angle B \cong \angle C \) is given). So the last blank should be \( \angle C \).

Answer:

1. The first blank: segment \( AC \)
2. The second part (theorem): If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side (Converse of Angle - Side Relationship)
3. The third blank: \( \angle C \)

(For the dropdowns in the original problem, the first dropdown should have "segment \( AC \)", and the last dropdown should have "angle \( C \)")