**Q) **Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contacts at the centre.

**Ans:**

Let’s draw a diagram with a circle of radius r and O as centre. Let the two tangents RP & RQ are drawn from point R on to this circle, The tangent RP touches the circle at point P and tangent RQ touches the circle at point Q.

We need to prove that, **∠** QRP + **∠** POQ = 180°

From the diagram, we can see that RQOP forms a quadrilateral and we know that, the sum of all 4 angles in any quadrilateral is 360^{0}.

**∠** QRP + **∠** RPO + **∠** POQ + **∠** RQO = 360° ……. (i)

Since we know that a tangent is always perpendicular to the radius, therefore

**∠** RPO = **∠** RQO = 90°

Substituting these values in equation (i), we get

**∠** QRP + 90° + **∠** POQ + 90° = 360°

**∠** QRP + **∠** POQ = 180° **…… Hence Proved!**