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



<Sorry, I cannot upload the figure. Kindly draw the figure, before analysing the answer>

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)
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...
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