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:

To determine the correct statement and reason, we analyze the parallel lines \(\overline{AB} \parallel \overline{DC}\) and the transversal \(\overline{BC}\). Alternate interior angles are congruent when lines are parallel. For \(\overline{AB} \parallel \overline{DC}\) with transversal \(\overline{BC}\), \(\angle ABC\) and \(\angle DCB\) are alternate interior angles. Let's check each option:

• First option: \(\angle ABC \cong \angle BAC\) is incorrect as they are not corresponding angles of parallel lines.
• Second option: \(\angle CAB\) and \(\angle DCB\) are not alternate interior angles for the given parallel lines.
• Third option: \(\angle ABC \cong \angle DCB\) (alternate interior angles of parallel lines are congruent) is correct.
• Fourth option: \(\angle ACD\) and \(\angle ABD\) are not corresponding angles of parallel lines.

Answer:

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