Question

given: \\( \angle b \cong \angle d \\) and \\( \overline{bc} \parallel \overline{ad} \\).
prove: \\( \overline{ab} \cong \overline{cd} \\).
note: quadrilateral properties are not permitted in the proof.

Explanation:

Step1: Find alternate - interior angles

Since \(\overline{BC}\parallel\overline{AD}\), by the Alternate - Interior Angles Theorem, \(\angle BCA\cong\angle DAC\).

Step2: Consider triangles \(\triangle ABC\) and \(\triangle CDA\)

We have \(\angle B\cong\angle D\) (given), \(\angle BCA\cong\angle DAC\) (from Step 1), and \(\overline{AC}\cong\overline{CA}\) (reflexive property of congruence).

Step3: Prove triangles congruent

By the Angle - Angle - Side (AAS) congruence criterion, \(\triangle ABC\cong\triangle CDA\).

Step4: Conclude the result

Since \(\triangle ABC\cong\triangle CDA\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\overline{AB}\cong\overline{CD}\).

Answer:

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