Question

given: \\( \angle b \cong \angle d \\) and \\( \overline{bc} \parallel \overline{ad} \\).
prove: \\( \overline{ab} \cong \overline{cd} \\).

step	statement	reason
	\\( \overline{bc} \parallel \overline{ad} \\)	given

Explanation:

Step1: Alternate interior angles

Since \( \overline{BC}\parallel\overline{AD}\), then \( \angle BCA\cong\angle DAC\) (alternate - interior angles theorem).

Step2: Congruent triangles

We have \( \angle B\cong\angle D\), \( \angle BCA\cong\angle DAC\), and \( AC = CA\) (reflexive property). By the AAS (Angle - Angle - Side) congruence criterion, \( \triangle ABC\cong\triangle CDA\).

Step3: Corresponding parts

If \( \triangle ABC\cong\triangle CDA\), then \( \overline{AB}\cong\overline{CD}\) (corresponding parts of congruent triangles are congruent).

Answer:

\(\overline{AB}\cong\overline{CD}\)