Question

given: δabc is isosceles; \\(\overline{ab} \cong \overline{ac}\\)
prove: \\(\angle b \cong \angle c\\)
we are given that δabc is isosceles with \\(\overline{ab} \cong \overline{ac}\\).
using the definition of congruent line segments, we know that dropdown.
lets assume that angles b and c are not congruent. then one angle measure must be greater than the other. if \\(m\angle b\\) is greater than \\(m\angle c\\), then ac is greater than ab by the dropdown.
however, this contradicts the given information that dropdown. therefore, dropdown, which is what we wished to prove.
similarly, if \\(m\angle b\\) is less than \\(m\angle c\\), we would reach the contradiction that \\(ab > ac\\). therefore, the angles must be congruent.

Explanation:

Step1: Fill congruent segments definition

Congruent segments have equal lengths: $AB = AC$.

Step2: Identify inequality theorem

If $m\angle B > m\angle C$, $AC > AB$ by Angle-Side Inequality Theorem.

Step3: State given contradiction

Contradicts given $\overline{AB} \cong \overline{AC}$ (or $AB = AC$).

Step4: Conclude angle congruence

Our assumption is false, so $\angle B \cong \angle C$.

Answer:

1. $AB = AC$
2. Angle-Side Inequality Theorem
3. $\overline{AB} \cong \overline{AC}$
4. $\angle B \cong \angle C$