Question

If the bisectors of angles of a quadrilateral enclose a rectangle, then show that it is a parallelogram.

Answer 1

Quadrilateral PQRS has angle bisectors $PT,QA,RA,SC.$

$\Delta PQB,\Delta QBT,\Delta SDC$ are right angled triangle.

Let angle $P=2x$

so, $\angle PQB=90-x=\angle BQT$

$\therefore \angle QTB=(90-(90-x\left)\right)=x$

$\angle CTR=180-x$

In triangle $SDR$,

$\angle RDS={90}^{\circ }$, in parallelogram $DCTR$

$\angle DCT$ & $\angle CDR={90}^{\circ }$

$\therefore \angle DRT=x$ & $\angle DRS=x$

$\therefore \angle DSR=90-x$

sum of adjacent angles, $\angle P+\angle Q={180}^{\circ }$

Opposite angles $\angle P=\angle R,\angle Q=\angle C$

$\therefore$ PQRS is parallelogram