Question

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

Answer 1

Hint: In this question, we will start with constructing the figure using given information. So, to construct the diagram we will start with making line XY, then it is produced to P, it is given that $ \angle XYZ = 64^\circ $ . A ray YQ will bisect the $ \angle ZYP $ , so it will divide $ \angle ZYP $ in two parts, i.e., $ \angle ZYQ = 58^\circ $ and $ \angle QYP = 58^\circ $ , then, afterwards we will find the value of $ \angle XYQ $ and reflex \[\angle QYP.\]

Complete step-by-step answer:
We need to construct a figure, for that we have been given few information in the question, which are $ \angle XYZ = 64^\circ $ , XY is produced to P and a ray YQ bisects $ \angle ZYP $ .
Using the above information, we will construct the figure below.

From the figure, we get that XYP is a straight line,
So, because of linear pair, s $ \angle XYZ + \angle ZYP = 180^\circ $
 $ \angle ZYP = 180^\circ - \angle XYZ $
Now, it is given that, $ \angle XYZ = 64^\circ $ \[.\] So, putting this in above equation, we get
 $
\Rightarrow \angle ZYP = 180^\circ - 64^\circ \\
\Rightarrow \angle ZYP = 116^\circ \\
  $
We have also been given that, YQ bisects $ \angle ZYP $
\[
\Rightarrow \angle ZYQ = \angle QYP = \dfrac{1}{2}\angle ZYP \\
   = \dfrac{1}{2} \times 116^\circ \\
   = 58^\circ \\
 \]
Now we need to find $ \angle XYQ $ \[,\]
So, $ \angle XYQ = \angle XYZ + \angle ZYQ $
 $
   = 64^\circ + 58^\circ \\
   = 122^\circ \\
  $
We also need to find reflex $ \angle QYP $ \[,\]
So, reflex $ \angle QYP = 360^\circ - \angle QYP $
 $
   = 360^\circ - 58^\circ \\
   = 302^\circ \\
  $
Thus,\[\angle XYQ = 122^\circ \] and reflex $ \angle QYP = 302^\circ $

Note: Students should carefully draw the diagram here using the given information, because from that only we will get the correct idea of the values. In the figure, there is a ray YQ is bisecting $ \angle ZYP $ , so it will divide $ \angle ZYP $ in two parts, i.e., $ \angle ZYQ = 58^\circ $ and $ \angle QYP = 58^\circ $ , because by angle bisector theorem, a line segment or ray divides angle into two equal parts.