Answer

Verified

393k+ views

**Hint:**We firstly need to find the angles $\angle 1$ and $\angle 2$. As we know that: The angle subtended by an arc of a circle at its center is twice the angle it subtends anywhere on the circle's circumference. So, we can say that: $\angle ROT=2\angle RST$. So, we get a value of $\angle 2$. Now, we know that $\angle POR={{130}^{\circ }}$. We get the value of $\angle POQ$ by supplementary relation. Now, in $\Delta POQ$ , we have, $\angle PQO={{90}^{\circ }}$ . We have found the value of $\angle POQ$. So, by using the sum of all angles of triangles relation, get the value of $\angle 1$. Now, add the values of $\angle 1$ and $\angle 2$ to get $\angle 1+\angle 2$

**Complete step-by-step solution**

We know that: The angle subtended by an arc of a circle at its centre is twice of the angle it subtends anywhere on the circle's circumference. So, we get:

$\angle ROT=2\angle RST$

Also, $\angle ROT=\angle POR={{130}^{{}^\circ }}$

So, we get:

\[\begin{align}

& \Rightarrow {{130}^{{}^\circ }}=2\angle RST \\

& \Rightarrow \angle RST={{65}^{{}^\circ }}......(1) \\

\end{align}\]

Therefore, $\angle 2={{65}^{{}^\circ }}$

Now, we know that the sum of all the angles on a line is ${{180}^{\circ }}$ . So, we get:

$\begin{align}

& \Rightarrow \angle ROT+\angle QOT={{180}^{{}^\circ }} \\

& \Rightarrow {{130}^{{}^\circ }}+\angle QOT={{180}^{{}^\circ }} \\

& \Rightarrow \angle QOT={{50}^{{}^\circ }}......(2) \\

\end{align}$

Now, in $\Delta POQ$ , we have,

$\angle PQO={{90}^{\circ }}$ (angle subtended by a tangent at a circle)

$\angle QOT={{50}^{{}^\circ }}$

So, we get:

$\begin{align}

& \Rightarrow \angle QOT+\angle PQO+\angle OPQ={{180}^{{}^\circ }} \\

& \Rightarrow {{50}^{{}^\circ }}+{{90}^{{}^\circ }}+\angle 1={{180}^{{}^\circ }} \\

& \Rightarrow \angle 1={{180}^{{}^\circ }}-{{140}^{{}^\circ }} \\

& \Rightarrow \angle 1={{40}^{{}^\circ }} \\

\end{align}$

Now, we need to find $\angle 1+\angle 2$

We have:

$\angle 1={{40}^{{}^\circ }}$

$\angle 2={{65}^{{}^\circ }}$

So,

$\begin{align}

& \angle 1+\angle 2={{40}^{{}^\circ }}+{{65}^{{}^\circ }} \\

& ={{105}^{{}^\circ }}

\end{align}$

**Note:**Important theorems included in this question are:

1. The angle formed at the center of the circle by lines originating from two points on the circle's circumference is double the angle formed on the circumference of the circle by lines originating from the same points.

2. Tangent to a circle is always perpendicular to the radius.

3. the sum of angles of a triangle equals the straight angle, i.e. ${{180}^{\circ }}$

4. Angles on a straight line add up to ${{180}^{\circ }}$

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Name 10 Living and Non living things class 9 biology CBSE

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

Write a letter to the principal requesting him to grant class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE