Question

In the figure below, lines XY and MN intersect at point O. If $\angle$POY = ${90}^{\circ }$ and $\angle$ A : $\angle$ B = 2 : 3, find $\frac{\left(\angle C\right)}{2}$.

• A. ${55}^{\circ }$
• B. ${68}^{\circ }$
• C. ${77.5}^{\circ }$
• D. ${63}^{\circ }$

Answer 1

The correct option is D ${63}^{\circ }$

Given, $\angle$POY = ${90}^{\circ }$ and
$\angle$A:$\angle$B = 2 : 3.
Let $\angle$A = 2x and $\angle$B = 3x.
Hence, $\angle$A + $\angle$B + $\angle$POY = ${180}^{\circ }$ (∵ XOY is a straight line)
$\Rightarrow$ 2x + 3x + ${90}^{\circ }$ = ${180}^{\circ }$
$\Rightarrow$ x = $\frac{90}{5}$ = ${18}^{\circ }$
∴ A = ${36}^{\circ }$ , B = ${54}^{\circ }$.
Also, MON is a straight line, so,
$\angle$B +$\angle$ C = ${180}^{\circ }$
$\Rightarrow$ ${54}^{\circ }$ + $\angle$ C = ${180}^{\circ }$
$\Rightarrow$ $\angle$C = ${180}^{\circ }$ – ${54}^{\circ }$ = ${126}^{\circ }$
$\Rightarrow$ $\frac{\left(\angle C\right)}{2}$ = ${63}^{\circ }$.