Question

In the given figure, PQRS is a cyclic Quadrilateral. Find measure of each of its angles.

• A.
  $\angle P={130}^{\circ }\angle R={50}^{\circ }$
  $\angle Q={30}^{\circ }\angle S={150}^{\circ }$
• B.
  $\angle P={135}^{\circ }\angle R={45}^{\circ }$
  $\angle Q={30}^{\circ }\angle S={150}^{\circ }$
• C.
  $\angle P={135}^{\circ }\angle R={45}^{\circ }$
  $\angle Q={35}^{\circ }\angle S={145}^{\circ }$
• D.
  $\angle P={130}^{\circ }\angle R={50}^{\circ }$
  $\angle Q={35}^{\circ }\angle S={145}^{\circ }$

Answer 1

The correct option is B

$\angle P={135}^{\circ }\angle R={45}^{\circ }$
$\angle Q={30}^{\circ }\angle S={150}^{\circ }$
Since, PQRS is a cyclic quadrilateral.

$\therefore \angle P+\angle R={180}^{\circ }$
and $\angle S+\angle Q={180}^{\circ }[\because \text{Sum of either pair of opposite angles}\text{is equal to}{180}^{\circ }]$

Substituting the values of given angles we get:
$\begin{array}{cc}3x+x& ={180}^{\circ }\\ \Rightarrow 4x& ={180}^{\circ }\\ \Rightarrow x& ={45}^{\circ }\end{array}$

$\begin{array}{cc}5y+y& ={180}^{\circ }\\ \Rightarrow y& ={30}^{\circ }\end{array}$

Substituting the values of x and y we get the angles as:

$\angle P=3x=3\times {45}^{\circ }={135}^{\circ }\angle Q=y={30}^{\circ }\angle R=x={45}^{\circ }\angle S=5y=5\times {30}^{\circ }={150}^{\circ }$

Hence, the correct answer is option (b).