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Question 8
If APB and COD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form
(A) a square
(B) a rhombus
(C) a rectangle
(D) any other parallelogram

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Solution

Given, APB and CQD are two parallel lines.

Let the bisectors of angles APQ and CQP meet at a point M and bisector 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=2NQP
[Since, PM and NQ are the angle bisectors of APQ and DQP respectively]
MPQ=NQP [Dividing both sides by 2]
{since, alternate interior angles are equal.]
PMQN
Similarly , BPQ=CQP [alternate interior angles]
PN ∥ QM
So, quadrilateral PMQN is a parallelogram.
CQD=180 [Since , CQD is a line]
CQP+DQP=1802MQP+2NQP=180
[Since , MQ and NQ are the bisectors of the angles CQP and DQP]
2(MCQ+NQP)=180MQN=90
Hence, PMQN is a rectangle.


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