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# In figure, lines XY and MN intersect at O. If $$\angle{POY}$$ = $$90^\circ$$ and a :b = 2:3. Find c.

It is given that, $$\angle{POY}$$ = $$90^\circ$$

Now, by Linear pair axiom,
$$\angle{POY}$$ + $$\angle{POX}$$ = $$180^\circ$$
$$\Rightarrow$$ $$\angle{POX}$$ = $$180^\circ$$ - $$\angle{POY}$$
$$\Rightarrow$$ $$\angle{POX}$$ = $$90^\circ$$
So, we get, a + b = $$90^\circ$$ ....(i)
It is also given that,
a :b = 2:3

Let a = 2k and b = 3k,
From eq. (i),
2k + 3k = $$90^\circ$$
$$\Rightarrow$$ 5k = $$90^\circ$$
$$\Rightarrow$$ k = $$18^\circ$$

So, a = 2 × $$18^\circ$$ = $$36^\circ$$
and b = 3 × $$18^\circ$$ = $$54^\circ$$
Again by Linear pair axiom,
$$\angle{MOX}$$ + $$\angle{XON}$$ = $$180^\circ$$
$$\Rightarrow$$ b + c = $$180^\circ$$
$$\Rightarrow$$ c = $$180^\circ$$ - $$54^\circ$$
$$\Rightarrow$$ c = $$126^\circ$$