Q. 74.3( 9 Votes )

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

Answer :


Sum of all angles of a quadrilateral is 360°


A + B + C + D = 360°


On dividing both sides by 2,


1/2(A + B + C + D) = 1/2 × 360° = 180°


AP, PB, RC and RD are bisectors of A, B, C and D


PAB + ABB + RCD + RDC = 180° …(1)


Sum of all angles of a triangle is 180°


∴ ∠PAB + APB + ABP = 180°


PAB + ABP = 180° – APB …(2)


Similarly,


∴ ∠RDC + RCD + CRD = 180°


RDC + RCD = 180° – CRD …(3)


Putting (2) and (3) in (1),


180° – APB + 180° – CRD = 180°


360° – APB CRD = 180°


APB + CRD = 360° – 180°


APB + CRD = 180° …(4)


Now,


SPQ = APB [vertically opposite angles]


SRQ = DRC [vertically opposite angles]


Putting in (4),


⇒ ∠SPQ + SRQ = 180°


Hence, PQRS is a quadrilateral whose opposite angles are supplementary.

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