Question

If TP and TQ are two tangents to a circle with centre O from an external point T so that $\angle\text{POQ} = 120^\circ$, then $\angle\text{PTQ}$ is equal to :

• A. $60^\circ$
• B. $70^\circ$
• C. $80^\circ$
• D. $90^\circ$

Hint:

Keep in mind that the angle between two tangents drawn from an external point to a circle is always supplementary to the angle subtended by the line segments joining the points of contact at the centre.
\[ \angle\text{PTQ} = 180^\circ - \angle\text{POQ} \]
Just subtract the given angle from $180^\circ$ to find the answer instantly: $180^\circ - 120^\circ = 60^\circ$.

Answer 1

The Correct Option is A

Step 1: Use the direct supplementary-angle property.
For two tangents drawn from an external point, the angle between the tangents and the angle subtended by the points of contact at the centre always add up to $180^\circ$: $\angle PTQ+\angle POQ=180^\circ$.
Step 2: Substitute the given angle.
$\angle PTQ + 120^\circ = 180^\circ$.
Step 3: Solve.
$\angle PTQ = 180^\circ-120^\circ = 60^\circ$.
\[ \boxed{60^\circ} \]