Question

In the alongside diagram, ABCD is a parallelogram in which AP bisects angle A and BQ bisects angle B.

State whether true or false for the following equation

PQ = CD

• A. True
• B. False

Answer 1

Answer: (A)

Let $O$ be the point of intersection of the angular bisectors $AP,andQB$

We know that,

1) Angular bisectors intersect at ${90}^{\circ }$.

2) Angular bisectors bisect each other, i.e., $AO=OP$ and $BO=OQ$

Join QP, to form $△POQ$

Consider $△AOB$ and $△POQ$,

$AO=OP$

$\angle AOB=\angle POQ={90}^{\circ }$

$BO=OQ$

The two triangles are congruent. Hence the side $AB$ and $QP$ must be equal. Hence, parallel.

We know that, in parallelogram $ABCD$,

$AB=CD$ and $AB\parallel CD$

Hence, $CD=QP$ and $CD\parallel QP$. Proved.