A transversal cuts two parallel lines at A and B, the two interior angles at A are bisected and so - Maths - Rectilinear Figures

Question

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

Decoder_8 · Accepted Answer

Dear student

To prove: ABCD is a rectangle
Proof:
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

Regards

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