8. Two parallel lines l andm are intersected bya transversalp (see Fig.)Show that thequadrilateral formedby the bisectors of interior anglesrectangle.

8. Two parallel lines l andm are intersected bya transversalp (see Fig

1.	8. Two parallel lines l andm are intersected bya transversalp (see Fig.)Show that thequadrilateral formedby the bisectors of interior anglesrectangle.
Answer» Given l\|\|m and p is the transversalTo prove: PQRS is a rectangleProof:RS, PS, PQ and RQ are bisectors of interior angles formed by the transversal with the parallel lines.∠RSP = ∠RPQ (Alternate angles)Hence RS\|\|PQSimilarly, 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.