Question

The quadrilateral formed by joining the angle bisectors of a cyclic quadrilateral is a

• A. Rectangle
• B. square
• C. parallelogram
• D. cyclic quadrilateral

Answer 1

The correct option is D

cyclic quadrilateral

ABCD is a cyclic quadrilateral ∴$\angle$A +$\angle$C = ${180}^{\circ }$ and $\angle$B+ $\angle$D = ${180}^{\circ }$

$\frac{1}{2}$$\angle$A+$\frac{1}{2}$ $\angle$C = ${90}^{\circ }$ and $\frac{1}{2}$ $\angle$B+$\frac{1}{2}$ $\angle$D = ${90}^{\circ }$

x + z = ${90}^{\circ }$ and y + w = ${90}^{\circ }$

In $△$ARB and $△$CPD, x+y + $\angle$ARB = ${180}^{\circ }$ and z+w+ $\angle$CPD = ${180}^{\circ }$

$\angle$ARB = ${180}^{\circ }$ – (x+y) and $\angle$CPD = ${180}^{\circ }$ – (z+w)

$\angle$ARB+$\angle$CPD = ${360}^{\circ }$ – (x+y+z+w) = ${360}^{\circ }$ – (90+90)

= ${360}^{\circ }$ – ${180}^{\circ }$ $\angle$ARB+$\angle$CPD = ${180}^{\circ }$

$\angle$SRQ+$\angle$QPS = ${180}^{\circ }$

The sum of a pair of opposite angles of a quadrilateral PQRS is ${180}^{\circ }$.

Hence PQRS is cyclic