Question

It is given that $\angle XYZ={64}^{}$ 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 $\angle QYP$.

Answer 1

Given,
$\angle XYZ={64}^{\circ }$

$YQ$ bisects $\angle ZYP$

$\angle XYZ+\angle ZYP={180}^{\circ }$(Linear Pair)

$\Rightarrow {64}^{\circ }+\angle ZYP={180}^{\circ }$

$\angle ZYP={116}^{\circ }$

Also, $\angle ZYP=\angle ZYQ+\angle QYP$

$\angle ZYQ=\angle QYP$ ($YQ$ bisects$\angle ZYP$)

$\Rightarrow \angle ZYP=2\angle ZYQ$

$\Rightarrow 2\angle ZYQ={116}^{\circ }$

$\Rightarrow \angle ZYQ={58}^{\circ }=\angle QYP$

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

$\Rightarrow \angle XYQ={64}^{\circ }+{58}^{\circ }$

$\Rightarrow \angle XYQ={122}^{\circ }$

Also, reflex $\angle QYP={180}^{\circ }+\angle XYQ$

$\angle QYP={180}^{\circ }+{122}^{\circ }$

$\Rightarrow \angle QYP={302}^{\circ }$