Question

The lines XY and MN intersect each other at point O as shown in the figure. If $\angle POY={90}^{\circ }$ and $a:b=2:3$, then find the value of $c$.

• A. 130${}^{\circ }$
• B. 128${}^{\circ }$
• C. 126${}^{\circ }$
• D. 124${}^{\circ }$

Answer 1

The correct option is C 126${}^{\circ }$

Given: $\angle POY={90}^{\circ }$
$a:b=2:3$

From the figure,
$\angle XOP$ and $\angle POY$ form a linear pair of angles.
$\Rightarrow a+b+\angle POY={180}^{\circ }$
$\Rightarrow a+b+{90}^{\circ }={180}^{\circ }$
$\Rightarrow a+b={90}^{\circ }$

Now, a:b = 2:3 (Given)
Let a and b be 2x and 3x.
$\Rightarrow$ $2x+3x={90}^{\circ }$
$\Rightarrow$ $5x={90}^{\circ }$
$\Rightarrow$ $x={18}^{\circ }$
Hence, $a=2\times {18}^{\circ }={36}^{\circ }$

$\angle MOY=a+\angle POY$
$\Rightarrow \angle MOY={36}^{\circ }+{90}^{\circ }={126}^{\circ }$
$\angle MOY=c$ ($\because$ both are vertically opposite angles)
$\therefore c=126°$

Alternative Method

$b=3\times {18}^{\circ }={54}^{\circ }$
$\angle MOX$ and $\angle XON$ form a linear pair of angles.
$\Rightarrow b+c={180}^{\circ }$
$\Rightarrow c={180}^{\circ }-{54}^{\circ }={126}^{\circ }$