Question

figure abcd is a parallelogram.
which sequence could be used to prove that ( overline{ad}=overline{bc} )?
first prove ( \triangle abc ) is congruent to ( \triangle cda ), and then state ( overline{ad} ) and ( overline{bc} ) are corresponding sides of the triangles.
first prove ( parallelogram abcd ) is congruent to ( parallelogram cdab ), and then state ( overline{ad} ) and ( overline{bc} ) are corresponding sides of two parallelograms.
first prove ( parallelogram abcd ) is similar to ( parallelogram cdab ), and then state ( overline{ad} ) and ( overline{bc} ) are opposite sides of the parallelograms.
first prove ( \triangle abc ) is similar to ( \triangle cda ), and then state ( overline{ad} ) and ( overline{bc} ) are opposite sides of the parallelograms.

Explanation:

Step1: Analyze parallelogram properties

In parallelogram \(ABCD\), \(AB\parallel CD\) and \(AD\parallel BC\). Also, \(\angle BAC=\angle DCA\) (alternate - interior angles) and \(\angle BCA = \angle DAC\) (alternate - interior angles), and \(AC = CA\) (common side).

Step2: Prove triangle congruence

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

Step3: Use congruent triangle properties

Since \(\triangle ABC\cong\triangle CDA\), corresponding parts of congruent triangles are equal. So, \(AD = BC\) as they are corresponding sides of \(\triangle ABC\) and \(\triangle CDA\).

Answer:

First prove \(\triangle ABC\) is congruent to \(\triangle CDA\), and then state \(\overline{AD}\) and \(\overline{BC}\) are corresponding sides of the triangles.