Question

figure abcd is a parallelogram which sequence could be used to prove that ad = bc? first prove △abc is congruent to △cda, and then state ad and bc are corresponding sides of the triangles first prove △abc is similar to △cda and then state ad and bc are opposite sides of the parallelograms

Explanation:

Step1: Properties of congruent triangles

If two triangles are congruent, their corresponding sides are equal.

Step2: Analyzing the first option

In parallelogram \(ABCD\), \(AB\parallel CD\) (so \(\angle BAC=\angle DCA\)) and \(AD\parallel BC\) (so \(\angle DAC = \angle BCA\)), and \(AC = AC\) (common side). By \(ASA\) (Angle - Side - Angle) criterion, \(\triangle ABC\cong\triangle CDA\). Then, since \(AD\) and \(BC\) are corresponding sides of congruent triangles \(\triangle ABC\) and \(\triangle CDA\), \(AD = BC\).

Step3: Analyzing the second option

Similar triangles have proportional sides, not necessarily equal sides. So proving \(\triangle ABC\sim\triangle CDA\) cannot directly give \(AD = BC\) as we need an equality of lengths (not just a proportion).

Answer:

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