Question

In the alongside diagram, $ABCD$ is a parallelogram in which $AP$ bisects $\angle A$ and $BQ$ bisects $\angle B$. Then, $ABPQ$ is a __________

• A. Square
• B. Rectangle
• C. Parallelogram
• D. Trapezium

Answer 1

The correct option is C Parallelogram

$ABPQ$ is a parallelogram

$ABCD$ is a parallelogram

$\Rightarrow AB||CD$ and $BC||AD$ [opposite sides are parallel to each other]

$AB=CD$ and $BC=AD$ [opposite sides are equal]

$\Rightarrow AD||BC$ and $AP$ is transversal.

$\angle QAP=\angle APB$ [alternate interior angle]

$\therefore △APB,\angle QAP=\angle PAB=\angle APB$

$\Rightarrow AB=BP$

with $AD||BC$ and $BQ$ is transversal,

$\angle QBD=\angle AQB$ [alternate interior angle]

In $△QAB$,

$\angle AQB=\angle ABQ=\angle QBP$

$\Rightarrow AQ=AB$

$\therefore AQ=BP$

As, $AD=BC$

$AQ+QD=BP+PC\Rightarrow QD=DC$

$\frac{AQ}{QD}=\frac{BP}{PC}$

$\therefore PQ||AB||CD$

$\therefore ABPQ$ is a parallelogram