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 size.
$\Rightarrow$ 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, $\angle POQ$ = $\angle SOR$ $\text{(as they are Vertically Opposite Angles)}$
$\Rightarrow$ $y=\angle POQ$
$\Rightarrow$ $={180}^{\circ }-(\angle OPQ+\angle OQP)$
$\Rightarrow$ $={180}^{\circ }-{60}^{\circ }={120}^{\circ }$
The required sum is,
$x+y={60}^{\circ }+{120}^{\circ }={180}^{\circ }$