Question

In Fig. lines $XY$ and $MN$ intersect at $O$. If $\angle POY=90°$ and $a:b=2:3$, find $c$.

Answer 1

Step 1: Find a relationship between $a,b$

$XY$ is a straight line. Hence the sum of all angles on it is ${180}^{\circ }$

$\therefore$ $\angle XOM+\angle MOP+\angle POY={180}^{\circ }$

$\therefore$ $b+a+{90}^{\circ }={180}^{\circ }$

$\therefore$ $b+a={90}^{\circ }$ $...\left(i\right)$

Step 2: Find value of $a,b$

Let the constant of proportionality be $x$

For, $a:b=2:3$

$\Rightarrow$ $a=2x,b=3x$ $...\left(ii\right)$

From $\left(i\right),\left(ii\right)$

$2x+3x={90}^{\circ }$

$\Rightarrow$ $5x={90}^{\circ }$

$\Rightarrow$ $x={18}^{\circ }$

$\therefore$ $$\begin{gathered} a=2x={36}^{\circ } \\ b=3x={54}^{\circ } \end{gathered}$$

Step 3: Find the value of $c$

$MN$ is a straight line. Hence the sum of all angles on it is ${180}^{\circ }$

$\therefore$ $\angle XOM+\angle XON={180}^{\circ }$

$\therefore$ $b+c={180}^{\circ }$

$\therefore$ $c={180}^{\circ }-{54}^{\circ }$

$\therefore$ $c={126}^{\circ }$

Hence, the value of $c$ is ${126}^{\circ }$