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

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Given:

It is given that $\angle XYZ=64^o$, $XY$ is produced to point $P$ and ray $YQ$ bisects $\angle ZYP$.

To do:

We have to draw a figure from the given information and find $\angle XYQ$ and reflex $\angle QYP$.

Solution:

$XYP$ is a line.

Therefore,

$\angle XYZ+\angle ZYP=180^o$

$64^o+\angle ZYP=180^o$  (since $\angle XYZ=64^o$)

This implies,

$\angle ZYP=180^o-64^o$

$\angle ZYP=116^o$

Since,

$YQ$ bisects $\angle ZYP$

We get,

$\angle ZYQ=\angle QYP$

and also,

$\angle ZYP=2\angle QYP$

This implies,

$116^o=2\angle QYP$

$\frac{116^o}{2}=\angle QYP$

$58^o=\angle QYP$

That is,

$\angle QYP=58^o$

Therefore,

$\angle ZYQ=\angle QYP=58^o$

Similarly, we get,

$\angle XYQ=\angle XYZ+\angle ZYQ$

This implies,

$\angle XYQ=64^o+58^o$

$\angle XYQ=122^o$

Now let us find,

Reflex $\angle QYP$

$\angle QYP=180^o+\angle XYQ$  (since $\angle QYP$ is reflex of $\angle XYQ$)

We have $\angle XYQ$ by substituting we get,

$\angle QYP=180^o+122^o$

This implies,

$\angle QYP=302^o$.

Hence, $\angle QYP=302^o$.

Updated on 10-Oct-2022 13:40:31