Q. 54.5( 2 Votes )

In the picture, bisectors of adjacent angles of the quadrilateral ABCD intersect at P, Q, R, S.



Prove that PQRS is a cyclic quadrilateral.

Answer :

In the given figure,


ASD = PSR [Vertically opposite angles]…[1]


In ΔASD


ASD + ADS + DAS = 180° [By angle sum property]


Also, As AS and DS are angle bisectors, therefore


and



Using these and from equation [1] we have,



…[2]


Similarly,


…[3]


Adding [2] and [3]



Also, A + B + C + D = 360° [By angle sum property of quadrilateral]



PSR + PQR = 180°


PQRS is a cyclic quadrilateral.


[As in a cyclic quadrilateral, sum of any pair of opposite angles is 180°].


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