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# It is given that $$\angle{XYZ}$$ = $$64^\circ$$ and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects $$\angle{ZYP}$$ , find $$\angle{XYQ}$$ and reflex$$\angle{QYP}$$.

Given that, ray YQ bisects $$\angle{ZYP}$$

Thus, $$\angle{ZYQ}$$ = $$\angle{QYP}$$ = $$\frac{1}{2}$$ ($$\angle{ZYP}$$) ...(i)

By Linear pair axiom,
$$\angle{XYZ}$$ + $$\angle{ZYQ}$$ + $$\angle{QYP}$$ = $$180^\circ$$

But, it is given that,

$$\angle{XYZ}$$ = $$64^\circ$$ ...(ii)

Therefore, from (i) and (ii), we get,

$$\therefore$$ $$64^\circ$$ + $$\angle{ZYQ}$$ + $$\angle{ZYQ}$$ = $$180^\circ$$
$$\Rightarrow$$ 2$$\angle{ZYQ}$$ = $$180^\circ$$ - $$64^\circ$$
$$\Rightarrow$$ 2$$\angle{ZYQ}$$ = $$116^\circ$$
$$\Rightarrow$$ $$\angle{ZYQ}$$ = $$58^\circ$$

Now,
$$\because$$ $$\angle{XYQ}$$ = $$\angle{XYZ}$$ + $$\angle{ZYQ}$$
$$\Rightarrow$$ $$\angle{XYQ}$$ = $$64^\circ$$ + $$58^\circ$$ = $$122^\circ$$

Now,
$$\angle{QYP}$$ + reflex$$\angle{QYP}$$ = $$360^\circ$$
$$\Rightarrow$$ $$58^\circ$$ + reflex$$\angle{QYP}$$ = $$360^\circ$$
$$\Rightarrow$$ reflex$$\angle{QYP}$$ = $$302^\circ$$