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Question 7
If bisectors of A and B of a quadrilateral ABCD intersect each other at P, of B and C at Q , of C and D at R and of D and A at S, then PQRS is a
(A) rectangle
(B) rhombus
(C) parallelogram
(D) quadrilateral whose opposite angles are supplementary

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Solution

Given, ABCD is a quadrilateral and all angles bisectors form a quadrilateral PQRS.

We know that, sum of all angles in a quadrilateral is 360.
A+B+C+D=360.
On dividing both sides by 2 , we get
12(A+B+C+D)=3602PAB+PBA+RCD+RDC=180.
[Since, AP and PB are the bisectors of A and B respectively also RC and RD are the bisectors of C and D respectively]
Now, in ΔAPB.
PAB+ABP+BPA=180 ...(i)
[By angle sum property of a triangle]
RDC+DCR=180CRD
On substituting the value eqs. (ii) and (iii) in Eq. (i) we get
180BPA+180DRC=180
BPA+180DRC=180
SPQ+SRQ=180[BPA=SPQ and DRC=SRQ vertically opposite angles]
Hence, PQRS is a quadrilateral whose opposite angle are supplementary.

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