In the given figure, $P Q$ is tangent to the circle centred at $O$. If $\angle A O B=95^{\circ}$, th

Question

In the given figure, $P Q$ is tangent to the circle centred at $O$. If $\angle A O B=95^{\circ}$, then the measure of $\angle A B Q$ will be\

Professor PC Thomas Classes · Answer

(a): We have $\angle A O B=95^{\circ}$ In $	riangle A O B, \angle O A B=\angle O B A$ (since $O A=O B$ ) Now, $\angle O A B+95^{\circ}+\angle O B A=180^{\circ}$\ (Angle sum property of a triangle)\ $\Rightarrow \quad \angle \mathrm{OAB}=\frac{85^{\circ}}{2}=42.5^{\circ}$\ $	herefore \quad \angle \mathrm{OAB}=\angle \mathrm{OBA}=42.5^{\circ}$ (From (i))\ Now, $O B$ is perpendicular to the tangent $P Q$.\ $	herefore \angle Q B O=90^{\circ} \Rightarrow \angle A B O+\angle A B Q=90^{\circ}$\ $\Rightarrow 42.5^{\circ}+\angle A B Q=90^{\circ} \Rightarrow \angle A B Q=47.5^{\circ}$