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# In figure, $$\angle{X}$$ = $$60^\circ$$ , $$\angle{XYZ}$$ = $$54^\circ$$ , if YO and ZO are the bisectors of $$\angle{XYZ}$$ and $$\angle{XZY}$$ respectively of $$\triangle{XYZ}$$ , find $$\angle{OZY}$$ and $$\angle{YOZ}$$.

In $$\triangle{XYZ}$$, $$\angle{X}$$ + $$\angle{Y}$$ + $$\angle{Z}$$ = $$180^\circ$$

because, Sum of all angles of triangle is equal to 180
$$\therefore$$ $$62^\circ$$ + $$\angle{Y}$$ + $$\angle{Z}$$ = $$180^\circ$$
$$\Rightarrow$$ $$\angle{Y}$$ + $$\angle{Z}$$ = $$118^\circ$$

Now, so as to find bisected angles, multiply both sides by $$\frac{1}{2}$$ We get,
$$\frac{1}{2}$$ [$$\angle{Y}$$ + $$\angle{Z}$$] = $$\frac{1}{2}$$ × $$118^\circ$$ = $$59^\circ$$

Thus we get,
$$\angle{OYZ}$$ + $$\angle{OZY}$$ = $$59^\circ$$
....(as YO and ZO are the bisectors of $$\angle{XYZ}$$ and $$\angle{XZY}$$)

Also, it is given that, $$\angle{XYZ}$$ = $$54^\circ$$ and we have
$$\Rightarrow$$ $$\angle{OYZ}$$ = $$\frac{1}{2}$$ × $$\angle{XYZ}$$
$$\Rightarrow$$ $$\angle{OZY}$$ + $$\frac{1}{2}$$ × $$54^\circ$$ = $$59^\circ$$
$$\Rightarrow$$ $$\angle{OZY}$$ = $$59^\circ$$ - $$27^\circ$$ = $$32^\circ$$

Also, in $$\triangle{YOZ}$$,

$$\angle{OYZ}$$ + $$\angle{YOZ}$$ + $$\angle{OZY}$$ = $$180^\circ$$

because, Sum of all angles of triangle is equal to 180
$$\therefore$$ $$27^\circ$$ + $$\angle{YOZ}$$ + $$32^\circ$$ = $$180^\circ$$
$$\Rightarrow$$ $$\angle{YOZ}$$ + $$59^\circ$$ = $$180^\circ$$
$$\Rightarrow$$ $$\angle{YOZ}$$ = $$180^\circ$$ - $$59^\circ$$
$$\Rightarrow$$ $$\angle{YOZ}$$ = $$121^\circ$$