Question

It is given that \( ∠\)XYZ = 64° and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects \(∠\)ZYP, find \(∠\)XYQ and reflex \(∠\)QYP.

Answer 1

Given: 

\[ \angle XYZ = 64^\circ \quad \text{and ray} \, YQ \, \text{bisects} \, \angle PYZ. \]

To Find:

\[ \angle XYQ \quad \text{and reflex} \, \angle QYP. \]

Since \( XY \) is produced to point \( P \), we have the straight line \( PX \) with rays \( YQ \) and \( YZ \) standing on it.

Hence, we can use the linear pair property:

\[ \angle XYZ + \angle ZYP = 180^\circ \]

Substituting \( \angle XYZ = 64^\circ \):

\[ 64^\circ + \angle ZYP = 180^\circ \] \[ \therefore \angle ZYP = 116^\circ \]

As \( YQ \) bisects \( \angle ZYP \), we have:

\[ \angle ZYQ = \angle QYP \] \[ \angle ZYP = 2 \times \angle ZYQ \] \[ \therefore \angle ZYQ = \angle QYP = 58^\circ \]

Now, to find \( \angle XYQ \):

\[ \angle XYQ = \angle XYZ + \angle ZYQ \] \[ \angle XYQ = 64^\circ + 58^\circ = 122^\circ \]

Finally, to find the reflex angle \( \angle QYP \):

\[ \text{Reflex} \, \angle QYP = 180^\circ + \angle XYQ \] \[ \therefore \angle QYP = 180^\circ + 122^\circ = 302^\circ \]