CBSE Class 9 Answered

let ABCD be the quadrilateral.

let the angle bisectors of angles A  and B  meet at  P

 and 

that of C  and D meet at Q.

 so a quadrilateral is formed with P and R as two of its vertices.

 let the other two vertices be Q and S, so that the quadrilateral is  PQRS.

consider triangle DQC,

angle Q+(C/2)+(D/2)=180....(i) angle sum property.

similarly,

in triangle APB.

 angle P+(A/2)+(B/2)=180..(ii)

ading (i) and (ii) and using the fact that sum of the angles A,B,C,D is 360 degrees

 we get,

angle P+ angle Q=180

 so quadrilateral PQRS is  cyclic as opp angles are supplementary.