Question

two parallel lines l and m are intersect by a transversal p show the quadrilateral formed by the bisectors of interior angles is rectangle

Answer 1

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.