Question

If the angle between two radii of a circle is ${130}^{\circ }$, then the angle between the tangents at the endpoints of radii at their point of intersection is ____ .

Answer 1

We know tangents are $\perp$ to radius at point of contact.

$\therefore$ $\angle CAO=\angle CBO={90}^o$

Now in quadrilateral $ABC,$

$\Rightarrow$ $\angle 1+\angle 2+\angle 3+\angle 4={360}^o$ [ Sum of four angles of a quadrilateral is ${360}^o$

$\Rightarrow$ $\angle 1+{90}^o+{90}^o+{130}^o={360}^o$

$\Rightarrow$ $\angle 1+{310}^o={360}^o$

$\Rightarrow$ $\angle 1={360}^o-{310}^o$

$\therefore$ $\angle 1={50}^o$

$\therefore$ Required measure of an angle is ${50}^o.$