Question

given: (overline{ab} parallel overline{dc}) (angle a cong angle d) prove: (\triangle abc cong \triangle dcb) isabelle proves that the triangles are congruent by using the parallel lines to determine a second set of angles are congruent. what statement and reason could she have used? (\bigcirc) (angle abc cong angle bac); corresponding angles of parallel lines are congruent. (\bigcirc) (angle cab cong angle dcb); alternate interior angles of parallel lines are congruent. (\bigcirc) (angle abc cong angle dcb); alternate interior angles of parallel lines are congruent. (\bigcirc) (angle acd cong angle abd); corresponding angles of parallel lines are congruent.

Explanation:

1. Analyze each option:

• Option 1: $\angle ABC \cong \angle BAC$ is incorrect. Corresponding angles of parallel lines don't justify this; also, these angles are not related by parallel lines here.
• Option 2: $\angle CAB \cong \angle DCB$ is incorrect. $\angle CAB$ and $\angle DCB$ are not alternate interior angles for lines $\overline{AB} \parallel \overline{DC}$ with transversal.
• Option 3: Since $\overline{AB} \parallel \overline{DC}$ and transversal is $\overline{BC}$, $\angle ABC$ and $\angle DCB$ are alternate interior angles. By the alternate interior angles theorem, they are congruent. This is valid for proving triangle congruence (along with given $\angle A \cong \angle D$ and common side or other congruent parts).
• Option 4: $\angle ACD \cong \angle ABD$ is incorrect. These angles are not related by the parallel lines $\overline{AB} \parallel \overline{DC}$ as corresponding or alternate interior angles.

Answer:

C. $\angle ABC \cong \angle DCB$; alternate interior angles of parallel lines are congruent