Question

If two parallel lines are intersected by a transversal then the bisectors of the interior angles form a :

• A. rhombus
• B. parallelogram
• C. Square
• D. Rectangle

Answer 1

The correct option is D

Rectangle

Using the given data,

Two parallel lines AB and CD are intersected by a transversal L at P and R respectively
PQ, RQ, RS and PS are bisectors of $\angle$APR, $\angle$PRC, $\angle$PRD and $\angle$BPR respectively.

Since AB || CD and L is a transversal

$\angle$APR = $\angle$PRD ( alt. interior angles)

$\angle$APR = $\frac{1}{2}$ $\angle$PRD = $\angle$QPR = $\angle$PRS

these are alternate interior angles.

QP || RS. Similarly QR || PS.

PQRS is a parallelogram.

Also ray PR stands on AB

$\angle$APR + $\angle$BPR = 180${}^{\circ }$ ( linear pair)

$\angle$APR + (1/2) $\angle$BPR = 90${}^{\circ }$

$\angle$QPR + $\angle$SPR = 90${}^{\circ }$

$\angle$QPS = 90${}^{\circ }$

Therefore PQRS is a parallelograrn, one of whose angle is 90${}^{\circ }$.
Hence PORS satisfies all the properties of being a rectangle.

Hence PQRS is a rectangle