question_answer

The quadrilateral formed by angle bisectors of a cyclic quadrilateral \[ABCD\] is a

A)  rhombus                      

B)  parallelogram

C)  cyclic quadrilateral          

D)  trapezium

Correct Answer: C

Solution :

            \[\angle APB+\angle PAB+PBA={{180}^{o}}\] and       \[\angle CRD+\angle RCD+\angle RDC={{180}^{o}}\] \[\Rightarrow \]   \[\angle APB+\frac{1}{2}\angle A+\frac{1}{2}\angle B={{180}^{o}}\] and       \[\angle CRD+\frac{1}{2}\angle C+\frac{1}{2}\angle D={{180}^{o}}\] \[\Rightarrow \]\[\angle APB+\angle CRD+\frac{1}{2}(\angle A+\angle B+\angle C+\angle D)\]         \[={{360}^{o}}\] \[\Rightarrow \]   \[\angle APB+\angle CRD={{180}^{o}}\] Hence \[PQRS\] is a cyclic quadrilateral.