Question

If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form a ____________.

Answer 1

Given:
APB and CQD are two parallel lines

Let the bisectors of the angle APQ and CQP intersects at the point R and the bisectors of the angle BPQ and PQD intersects at the point S.

Join PR, RQ, QS and SP as shown in the figure.



APB || CQD
⇒ ∠APQ = ∠PQD (alternate angles)
⇒ 2∠RPQ = 2∠PQS
⇒ ∠RPQ = ∠PQS
⇒ RP || SQ ...(1)

Similarly,
APB || CQD
⇒ ∠BPQ = ∠PQC (alternate angles)
⇒ 2∠SPQ = 2∠PQR
⇒ ∠SPQ = ∠PQR
​⇒ RQ || SP ...(2)

From (1) and (2),
PSQR is a parallelogram,

Also, ∠CQP + ∠PQD = 180° (angles on a straight line)
⇒ 2∠RQP + 2∠PQS = 180°
⇒ ∠RQP + ∠PQS = 90°
⇒ ∠RQS = 90°

Therefore, PSQR is a rectangle.

Hence, if APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form a rectangle.