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प्रश्न
If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form ______.
विकल्प
a square
a rhombus
a rectangle
any other parallelogram
उत्तर
If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form a rectangle.
Explanation:
Given, APB and CQD are two parallel lines.
Let the bisectors of angles APQ and CQP meet at a point M and bisectors of angles BPQ and PQD meet at a point N.
Join PM, MQ, QN and NP.
Since, APB || CQD
Then, ∠APQ = ∠PQD ...[Alternate interior angles]
⇒ ∠MPQ = 2∠NQP ...[Since, PM and NQ are the angle bisectors of ∠APQ and ∠DQP respectively]
⇒ ∠MPQ = ∠NQP ...[Dividing both sides by 2] [Since, alternative interior angles are equal]
∴ PM || QN
Similarly, ∠BPQ = ∠CQP ...[Alternate interior angles]
∴ PN || QM
So, quadrilateral PMQN is a parallelogram.
∵ ∠CQD = 180° ...[Since, CQD is a line]
⇒ ∠CQP + ∠DQP = 180°
⇒ 2∠MQP + 2∠NQP = 180° ...[Since, MQ and NQ are the bisectors of the angles CQP and DQP]
⇒ 2(∠MQP + ∠NQP) = 180°
⇒ ∠MQN = 90°
Hence, PMQN is a rectangle.
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