Question

The angle-bisectors of a parallelogram form a quadrilateral as shown in the figure. What can be said about the quadrilateral ABCD?

• A. It's a parallelogram with three acute angles
• B. It's a parallelogram with three obtuse angles
• C. It's a parallelogram with each angle equal to ${90}^{\circ }$
• D. It's a parallelogram with adjacent sides equal

Answer 1

The correct option is C

It's a parallelogram with each angle equal to ${90}^{\circ }$

The quadrilateral formed is a rectangle.
Clearly, from the fig.,
$\angle$QPS + $\angle$PSR = ${180}^{\circ }$ (1)

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

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