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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