Question

In the given figure, PQRS is a rectangle if $\angle RPQ={30}^{\circ }$. Then, the value of $(x+y)$ is:

• A. ${90}^{\circ }$
• B. ${120}^{\circ }$
• C. ${150}^{\circ }$
• D. ${180}^{\circ }$

Answer 1

The correct option is D

${180}^{\circ }$

We know that, the diagonals of a rectangle bisect each other
$\Rightarrow$ OP = OQ = OR = OS

Given, $\angle RPQ=\angle OPQ={30}^{\circ }$
Since, $OP=OQ,$
$\angle OPQ=\angle OQP={30}^{\circ }$

All the angles of a rectangle are ${90}^{\circ }$.
$x={90}^{\circ }-\angle OQP={90}^{\circ }-{30}^{\circ }={60}^{\circ }$

Also, $\angle POQ=\angle SOR=y$ $\text{(Vertically Opposite Angles)}$
$\Rightarrow y={180}^{\circ }-(\angle OPQ+\angle OQP)$
$\Rightarrow y={180}^{\circ }-{60}^{\circ }={120}^{\circ }$

Hence, the required sum is,
$x+y={60}^{\circ }+{120}^{\circ }={180}^{\circ }$