Question

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

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 of congruent triangles

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

Answer:

$\overline{AB}\cong\overline{CD}$ is proved as above.