A transversal cuts two parallel lines at A and B, the two interior angles at A are bisected and so are the two interior angles at B, the four bisectors form a quadrilateral ACBD. Prove that ACBD is a rectangle and CD is parallel to the original parallel lines

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To prove: ABCD is a rectangle
CD, AD, AB and CB are bisectors of interior angles formed by the transversal with the parallel lines.
∠CDA = ∠CAB (Alternate angles)
Hence CD||AB
Similarly, AD||BC (∠CAD = ∠ACB)
Therefore quadrilateral ABCD 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 ABCD is a parallelogram and one of the angle is a right angle.
Hence ABCD is a rectangle.


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