Question

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

answer
you must answer all questions above in order to submit.

Explanation:

Step1: Use parallel line property

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

Step2: Use AAS congruence

In \(\triangle ABC\) and \(\triangle CDA\), we have \(\angle B=\angle D\) (given), \(\angle BCA = \angle DAC\) (from step 1), and \(AC = CA\) (common side). By the Angle - Angle - Side (AAS) congruence criterion, \(\triangle ABC\cong\triangle CDA\).

Step3: Use congruent triangle property

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

Answer:

\(\overline{AB}\cong\overline{CD}\) because \(\triangle ABC\cong\triangle CDA\) (by AAS) and corresponding parts of congruent triangles are congruent.