Question

given abcd is a parallelogram. prove ∠a≅∠c and ∠b≅∠d. by the definition of a ▱, ad∥bc and ab∥dc. using, ad as a transversal, ∠a and ∠ are same - side interior angles, so they are . using side as a transversal, ∠b and ∠c are same - side interior angles, so they are supplementary. using ab as a transversal, ∠a and ∠b are same - side interior angles, so they are supplementary. therefore, ∠a is congruent to ∠c because they are supplements of the same angle. similarly, ∠b is congruent to ∠

Explanation:

Step1: Recall properties of parallelogram

In parallelogram \(ABCD\), \(AD\parallel BC\) and \(AB\parallel DC\).

Step2: Analyze angles with \(AD\) as transversal

When \(AD\) is a transversal for \(AB\parallel DC\), \(\angle A\) and \(\angle D\) are same - side interior angles, so \(\angle A+\angle D = 180^{\circ}\) (supplementary).

Step3: Analyze angles with \(BC\) as transversal

When \(BC\) is a transversal for \(AB\parallel DC\), \(\angle B\) and \(\angle C\) are same - side interior angles, so \(\angle B+\angle C=180^{\circ}\) (supplementary).

Step4: Analyze angles with \(AB\) as transversal

When \(AB\) is a transversal for \(AD\parallel BC\), \(\angle A\) and \(\angle B\) are same - side interior angles, so \(\angle A+\angle B = 180^{\circ}\) (supplementary).

Step5: Prove angle - congruence

Since \(\angle A+\angle D = 180^{\circ}\) and \(\angle B+\angle D=180^{\circ}\), then \(\angle A\cong\angle C\) (because they are supplements of the same angle \(\angle D\)). Similarly, since \(\angle A+\angle B = 180^{\circ}\) and \(\angle C+\angle B = 180^{\circ}\), then \(\angle B\cong\angle D\).

Answer:

We have proved that \(\angle A\cong\angle C\) and \(\angle B\cong\angle D\) in parallelogram \(ABCD\).