Question

If bisectors of $\angle Aand\angle B$ of a quadrilateral ABCD intersect each other at P, of $\angle Band\angle C$ at Q, of $\angle Cand\angle D$ at R and of $\angle Dand\angle A$ at S then PQRS is a

(a) rectangle

(b) parallelogram

(c) rhombus

(d) quadrilateral whose opposite angles are supplementary

Answer 1

Given: In quadrilateral ABCD, AS, BQ, CQ and DS are
angle bisectors of angles A, B, C and D, respectively.
$\angle QPS=\angle APB$ (Vertically opposite angles)......(1)
In ∆APB,
$\angle APB+\angle PAB+\angle ABP={180}^{\circ }$
(Angle sum property of triangle.)
⇒ $\angle APB+\frac{1}{2}\angle A+\frac{1}{2}\angle B={180}^o$
⇒ $\angle APB={180}^{\circ }–\frac{1}{2}(\angle A+\angle B)$...(2)
From (1) and (2), we get
$\angle QPS={180}^{\circ }–\frac{1}{2}(\angle A+\angle B)$.......(3)
Similarly,
$\angle QRS={180}^{\circ }–\frac{1}{2}(\angle C+\angle D)$.......(4)
From (3) and (4), we get
$\angle QPS+\angle QRS={360}^{\circ }–\frac{1}{2}(\angle A+\angle B+\angle C+\angle D)$
=${360}^{\circ }–\frac{1}{2}(360{)}^{\circ }$
= ${360}^{\circ }–{180}^{\circ }$
=${180}^{\circ }$
So, PQRS is a quadrilateral whose opposite angles are supplementary.
Hence, the correct option is (d).