Question

prove: \\( \overline { a b } \cong \overline { c d } \\).
note: quadrilateral properties are not permitted in this proof.

Explanation:

Step1: Use alternate interior angles theorem

Since \(BC\parallel AD\), by the alternate - interior angles theorem, \(\angle BCA\cong\angle DAC\).

Step2: Prove triangles congruent

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

Step3: Use congruent triangle properties

Since \(\triangle ABC\cong\triangle CDA\), by the corresponding parts of congruent triangles (CPCTC), \(\overline{AB}\cong\overline{CD}\).

Answer:

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