Question

If the diagonals of a parllelogram are equal, then show that it is a rectangle.

Answer 1

Let say, $ABCD$ is a parallelogram

Given that the diagonals $AC$ and $BD$ of parallelogram

$ABCD$ are equal in length.

Consider triangles $ABC$ and $ACD$.

$AC=BD$ [Given]

$AB=AD$ [ common side]

$\therefore \Delta ABC\cong \Delta DCA$ [ SAS congruence criterion]

$\angle BAD=\angle CDA\left[CPCT\right]$

$\angle BAD=\angle CDA={180}^{\circ }$

[ Adjacent angles of a parallelogram are supplementary]

So, $\angle BAD$ and $\angle CDA$ are right angles as they are congruent and supplementary.

Therefore, parallelogram ABCD is a rectangle since a parallelogram with one right interior angle is a rectangle.

Hence, we proved.