Question

If two parallel lines are intersected by a transverse line, then the bisectors of the interior angles forms:

• A. a square
• B. a rectangle
• C. a parallelogram
• D. a trapezium

Answer 1

The correct option is B a rectangle

Given : $l||m$ and $p$ is the transversal
To prove: $PQRS$ is a rectangle
Proof:
$RS,PS,PQandRQ$ are bisectors of interior angles formed by the transversal with the parallel lines
$\angle RSP=\angle RPQ$ (Alternate angles)
Hence, $RS||PQ$
Similarly, $PS||RQ$ ($\angle RPS=\angle PRQ$)
$\therefore$ quadrilateral $PQRS$ is a parallelogram as both the pairs of opposite sides are parallel.
From the figure, we have $\angle b+\angle b+\angle a+\angle a={180}^o$
⇒ $2(\angle b+\angle a)={180}^o$
∴ $\angle b+\angle a={90}^o$
That is $PQRS$ is a parallelogram and one of the angle is a right angle.
Hence, $PQRS$ is a $Rectangle$