# Two parallel lines are interesected by a transversal P . Show that the quadrilateral showed by tbe interior angles is a rectangle

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Given

l||m

p is the transversal

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 (since ∠RPS = ∠PRQ)

Therefore, the 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 with an angle as a right angle.

Hence, PQRS is a rectangle.

Hope this helps...