Question

In the figure, find the measure of $\angle MPN$.

Answer 1

In the figure, OMPN is a quadrilateral in which $\angle O={45}^{\circ },\angle M=\angle N={90}^{\circ }$
$(\because PM\perp OAandPN\perp OB)$
Let $\angle MPN=x^{\circ }$
$\therefore \angle O+\angle M+\angle N+\angle MPN={360}^{\circ }$
(sum of angles of a quadrilateral)
$\Rightarrow {45}^{\circ }+{90}^{\circ }+{90}^{\circ }+x^{\circ }={360}^{\circ }\Rightarrow {225}^{\circ }+x^{\circ }={360}^{\circ }\Rightarrow x^{\circ }={360}^{\circ }-{225}^{\circ }\Rightarrow x={135}^{\circ }\therefore \angle MPN={135}^{\circ }$