Question

If each pair of opposite sides of a quadrilateral is equal the it is a parallelogram.

Answer 1

Given:- A quadrilateral $ABCD$ in which $AB=CD$ and $AD=BC$.

To prove:- $AB\parallel CD$ and $AD\parallel BC$

Construction:- Joi $A$ and $C$.

Proof:-

In $△ABC$ and $△CDA$,

$AB=CD\left(\text{Given}\right)$

$AD=BC\left(\text{Given}\right)$

$AC=AC\left(\text{Common}\right)$

$\therefore △ABC\cong △CDA\left(\text{By SSS}\right)$

Therefore, by CPCTC,

$\angle ACD=\angle BAC$

$\angle DAC=\angle ACB$

Since alternate anglea are equal, thus the ines are parallel.

Therefore,

$AB\parallel CD$ and $AD\parallel BC$

Since both the pairs of opposite sides of quadrilateral are parallel, $ABCD$ is a parallelogram.

Hence proved.