Question

given: ab || dc
∠a ≅ ∠d
prove: △abc ≅ △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?
∠abc ≅ ∠bac; corresponding angles of parallel lines are congruent.
∠cab ≅ ∠dcb; alternate interior angles of parallel lines are congruent
∠abc ≅ ∠dcb; alternate interior angles of parallel lines are congruent
∠acd ≅ ∠abd; corresponding angles of parallel lines are congruent.

Explanation:

When two parallel lines ($\overline{AB}\parallel\overline{DC}$) are cut by a transversal (in this case, $\overline{BC}$), the alternate - interior angles are congruent. $\angle ABC$ and $\angle DCB$ are alternate - interior angles formed by the parallel lines $\overline{AB}$ and $\overline{DC}$ with transversal $\overline{BC}$.

Answer:

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