Construction of Half Angles

Q. Prove that tangent drawn at any point of a circle is perpendicular to the radius through the point of contact.

Q.

In figure below P, Q, R are the points where the incircle of $\Delta ABC$ touches the sides. Find $\angle AQR.$

1. ${80}^{\circ }$

2. ${45}^{\circ }$

3. ${70}^{\circ }$

4. ${180}^{\circ }$

Q. In the given figure, $Q$ is the centre of the circle and $PM$ and $PN$ are tangent segments to the circle.
If $m\angle MPN={70}^o$, find $\angle MQN$.

Q.

In the given figure, O is the centre of the circle. Find $x$.

1. ${50}^{\circ }$

2. ${40}^{\circ }$

3. ${20}^{\circ }$

4. ${30}^{\circ }$