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

${180}^{\circ }$

The diagonals of a rectangle bisect each other and are equal in length.

$\Rightarrow OP=OQ=OR=OS$
Given, $\angle RPQ=\angle OPQ={30}^{\circ }$
Since, $OP=OQ\Rightarrow \angle OPQ=\angle OQP={30}^{\circ }$
All the angles of a rectangle are ${90}^{\circ }$.
$x={90}^{\circ }-\angle OQP={90}^{\circ }-{30}^{\circ }={60}^{\circ }$

Also, $\angle OQR$ = $\angle ORQ$ $(\because OR=OQ)$
$\Rightarrow \angle OQR=\angle ORQ=x$
Now, $y=\angle OQR+\angle ORQ=x+x$
$\because \text{( In a triangle, exterior angle is equal sum interior opposite Angles)}$
$\Rightarrow y=60+60={120}^0$
The required sum is,
$x+y={60}^{\circ }+{120}^{\circ }={180}^{\circ }$