# if l and m are intersected by p and q then prove that abcd is a //gm

To prove: PQRS is a rectangle

Proof:

RS, PS, PQ and RQ are bisectors of interior angles formed by the transversal with the parallel lines.

∠RSP = ∠RPQ (Alternate angles)

Hence RS||PQ

Similarly, PS||RQ (∠RPS = ∠PRQ)

Therefore quadrilateral PQRS is a parallelogram as both the pairs of opposite sides are parallel.

From the figure, we have ∠b + ∠b + ∠a + ∠a = 180°

⇒ 2(∠b + ∠a) = 180°

∴ ∠b + ∠a = 90°

That is PQRS is a parallelogram and one of the angle is a right angle.

Hence PQRS is a rectangle.