Answer

Verified

438.6k+ views

**Hint**: To solve this question, we have been given a figure, where, $\angle ROQ = 90^\circ $ and $\angle ROP = 90^\circ $ then, on keeping $\angle ROP$ and $\angle ROQ$ equal, because the angles are equals to \[90^\circ ,\] then solving further we will get the required expression.

**:**

__Complete step-by-step answer__We have been given a figure, in it POQ is a line. We have also been given that the ray OR is perpendicular to line PQ. And, OS is another ray lying between rays OP and OR. We need to prove that $\angle ROS = \dfrac{1}{2}(\angle QOS - \angle POS).$

So, it is given that ray OR is perpendicular to line PQ, i.e., $OR \bot PQ$

From the figure we get that, $\angle ROP = 90^\circ $ and $\angle ROQ = 90^\circ $

So, we can put, $\angle ROP = \angle ROQ$ (because both the angles are equals to \[90^\circ \])

Since, it is given that OS is another ray lying between rays OP and OR.

$ \Rightarrow \angle ROP = \angle POS + \angle ROS$

Now, on putting, $\angle ROQ$in place of $\angle ROP$, because both the angles are equal to each other, we get

$\angle POS + \angle ROS = \angle ROQ$

Since in the question, it is given that, ray OR is perpendicular to line PQ.

$ \Rightarrow \angle POS + \angle ROS = \angle QOS - \angle ROS$

Now, on rearranging the above equation, we get

$2\angle ROS = \angle QOS - \angle POS$

$ \Rightarrow \angle ROS = \dfrac{1}{2}\angle QOS - \angle POS$

Hence proved, $\angle ROS = \dfrac{1}{2}(\angle QOS - \angle POS).$

**Note**: Before solving, students should carefully observe the given figure. And then using the given information, solve further. To solve these types of question, where we are supposed to prove something, it is usually advisable to start with one side i.e., either start from L.H.S or R.H.S., then eventually we can prove the equation.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths