if l and m are intersected by p and q then prove that abcd is a //gm
Given l||m and p is the transversal
To prove: PQRS is a rectangle
RS, PS, PQ and RQ are bisectors of interior angles formed by the transversal with the parallel lines.
∠RSP = ∠RPQ (Alternate angles)
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.