The angle-bisectors of a parallelogram forms a quadrilateral as shown in the figure. Show that ABCD is a rectangle.

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Solution

Given quadrilateral PQRS is a paralleogram.
$∴$ $∠$ QPS + $∠$ PSR = $180^{\circ}$ (because consecutive angles of a parallelogram are supplementary) (1)

As, PC and SA are the respective angle bisectors of $∠$ P and $∠$ S,
on dividing both sides of (1) by 2, we have,
$∠$ DPS + $∠$ PSD = $90^{\circ}$

Thus, in $△$ PDS, $∠$ PDS = $90^{\circ}$
or, $∠$ ADC = $∠$ PDS = $90^{\circ}$ (vertically opposite angles)
Similarly, $∠$ DCB = $∠$ CBA = $∠$ BAD = $90^{\circ}$
Thus, fig.ABCD is a rectangle.

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