Question

In the diagram, $PQ$ is a tangent to the circle at $P$ and $QRS$ is a straight line. Then the value of $x$ is:

• A. ${20}^o$
• B. ${25}^o$
• C. ${30}^o$
• D. ${40}^o$

Answer 1

The correct option is B ${25}^o$

From the given figure,

$\angle TPS=\angle PSQ+\angle SQP$ ..... [exterior angle of a triangle is equal to interior opposite angle]
$65=x+40\Rightarrow x={25}^o$

(Alternate Method)

$\because m\left(\angle P\right)={65}^o$

$\therefore m\left(\angle SPQ\right)=180-65={115}^o$
Now in triangle $SPQ$,

Sum of all angles of $△SPQ$ will be equal to ${180}^o$
$\therefore$ $x+{115}^o+{40}^o={180}^o$
$⟹x={25}^o$

Hence, option B is correct.