Question

If $\angle POR={{120}^{\circ }}$ and P-O-L, the points S and T are on the R-side of line PL. Such that $\angle ROS\cong \angle SOL$ and $\angle ROT\cong \angle TOP$ . Draw the figure and find $\angle TOS$ .
$\begin{align}
  & \text{A}\text{. }\angle TOS={{90}^{\circ }} \\
 & \text{B}\text{. }\angle TOS={{70}^{\circ }} \\
 & \text{C}\text{. }\angle TOS={{400}^{\circ }} \\
 & \text{D}\text{. }\angle TOS={{120}^{\circ }} \\
\end{align}$

Answer 1

Hint: To find $\angle TOS$ first we will draw a diagram by the information given in the question. Then by taking $\angle POR+\angle ROS+\angle SOL={{180}^{\circ }}$ we will find the angles of ROS & SOL. Then we will find the angle of ROT by taking $\angle POR=\angle ROT+\angle TOP$ (from the diagram). At last we will find $\angle TOS$ by taking $\angle TOS=\angle ROT+\angle ROS$ .
Complete step by step solution:
Here, in this question we are asked to find the $\angle TOS$ . For that first we will draw the figure with the information given in the question.

From the figure we get to know that –
$\angle POR=120{}^\circ $ ,
PL is a straight line which is equal to $180{}^\circ $ ,
T and S are the points at the R-side of the line PL.
Let us consider $\angle ROS=x$ , $\angle SOL=x$ , $\angle ROT=y$ and $\angle TOP=y$
It is also given in the question that –
$\angle ROS=\angle SOL$ …………………. (1)
And $\angle ROT=\angle TOP$ …………………….. (2)
From the figure, we get that $PL=\angle POR+\angle ROS+\angle SOL$
We know that $PL=180{}^\circ $
So, $\angle POR+\angle ROS+\angle SOL=180{}^\circ $
By substituting their values, we get –
$\begin{align}
  & 120+x+x=180 \\
 & 120+2x=180 \\
\end{align}$
By subtracting 120 on both sides, we get –
$\begin{align}
  & 2x=180-120 \\
 & 2x=60 \\
\end{align}$
By dividing both sides by 2, we get –
$\begin{align}
  & x=\dfrac{60}{2} \\
 & x=30 \\
\end{align}$
Therefore, from the equation (1)
$\Rightarrow \angle ROS=\angle SOL=x=30{}^\circ $ …………………. (3)
Now, we will find the $\angle ROT$ .
From the figure we get that –
$\angle POR=\angle ROT+\angle TOP$
By substituting their values, we get –
$\begin{align}
  & 120=y+y \\
 & 120=2y \\
\end{align}$
By dividing both sides by 2, we get –
$y=\dfrac{120}{2}$
$y=60$
Therefore from the equation (2), we get –
$\Rightarrow ROT=\angle TOP=y=60{}^\circ $ ……………………… (4)
Now, we will find $\angle TOS$
From the figure, we get that –
$\angle TOS=\angle ROT+\angle ROS$
By substituting the values from equation (3) and (4), we get –
$\begin{align}
  & \angle TOS=60+30 \\
 & \therefore \angle TOS={{90}^{\circ }} \\
\end{align}$
Hence, option (A) is the correct answer.

Note: Generally students do mistake while drawing a diagram by marking the point as T instead of S as given below –

According to this figure $\angle ROS\ne \angle SOL$ and $\angle ROT\ne \angle TOP$which is totally different then the information given in the question. And this simple mistake can lead them to get the wrong answer. So students need to be very careful while solving drawing such diagrams according to the info given in the question.