Question

Question 6
It is given that $\angle XYZ={64}^{\circ }$ and XY is produced to point P. Draw a figure from the given information. If a 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)
$\Rightarrow {64}^{\circ }+\angle ZYP={180}^{\circ }$
$\Rightarrow \angle ZYP={116}^{\circ }$
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,
$reflex\angle QYP={180}^{\circ }+\angle XYQ$
$reflex\angle QYP={180}^{\circ }+{122}^{\circ }$
$={302}^{\circ }$