Question

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

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 $\mathrm{\angle }XYZ={64}^{\circ }$ . A ray YQ will bisect the $\mathrm{\angle }ZYP$ , so it will divide $\mathrm{\angle }ZYP$ in two parts, i.e., $\mathrm{\angle }ZYQ={58}^{\circ }$ and $\mathrm{\angle }QYP={58}^{\circ }$ , then, afterwards we will find the value of $\mathrm{\angle }XYQ$ and reflex $$\mathrm{\angle }QYP.$$

Complete step-by-step answer:
We need to construct a figure, for that we have been given few information in the question, which are $\mathrm{\angle }XYZ={64}^{\circ }$ , XY is produced to P and a ray YQ bisects $\mathrm{\angle }ZYP$ .
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 $\mathrm{\angle }XYZ+\mathrm{\angle }ZYP={180}^{\circ }$
 $\mathrm{\angle }ZYP={180}^{\circ }-\mathrm{\angle }XYZ$
Now, it is given that, $\mathrm{\angle }XYZ={64}^{\circ }$ $$.$$ So, putting this in above equation, we get
 $$\begin{gathered} \Rightarrow \mathrm{\angle }ZYP={180}^{\circ }-{64}^{\circ } \\ \Rightarrow \mathrm{\angle }ZYP={116}^{\circ } \end{gathered}$$
We have also been given that, YQ bisects $\mathrm{\angle }ZYP$
$$\begin{gathered} \Rightarrow \mathrm{\angle }ZYQ=\mathrm{\angle }QYP={\displaystyle \frac{1}{2}}\mathrm{\angle }ZYP \\ ={\displaystyle \frac{1}{2}}\times {116}^{\circ } \\ ={58}^{\circ } \end{gathered}$$
Now we need to find $\mathrm{\angle }XYQ$ $$,$$
So, $\mathrm{\angle }XYQ=\mathrm{\angle }XYZ+\mathrm{\angle }ZYQ$
 $$\begin{gathered} ={64}^{\circ }+{58}^{\circ } \\ ={122}^{\circ } \end{gathered}$$
We also need to find reflex $\mathrm{\angle }QYP$ $$,$$
So, reflex $\mathrm{\angle }QYP={360}^{\circ }-\mathrm{\angle }QYP$
 $$\begin{gathered} ={360}^{\circ }-{58}^{\circ } \\ ={302}^{\circ } \end{gathered}$$
Thus,$$\mathrm{\angle }XYQ={122}^{\circ }$$ and reflex $\mathrm{\angle }QYP={302}^{\circ }$

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 $\mathrm{\angle }ZYP$ , so it will divide $\mathrm{\angle }ZYP$ in two parts, i.e., $\mathrm{\angle }ZYQ={58}^{\circ }$ and $\mathrm{\angle }QYP={58}^{\circ }$ , because by angle bisector theorem, a line segment or ray divides angle into two equal parts.