In the given figure, if TP and TQ are tangents to a circle with centre O, so that `angle POQ= 110^(@)`, then `angle PTQ` is

Options

A. `110^(@)`
B. `90^(@)`
C. `80^(@)`
D. `70^(@)`

Explanation

To solve the problem, we need to find the measure of angle PTQ given that angle POQ is 110 degrees and TP and TQ are tangents to the circle with center O.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We have a circle with center O.
- TP and TQ are tangents to the circle at points P and Q respectively.
- Angle POQ = 110 degrees.

2. **Use the Property of Tangents:**
- The radius of the circle at the point of tangency is perpendicular to the tangent. Therefore, we have:
- Angle OPT = 90 degrees (since OP is the radius and TP is the tangent).
- Angle OQT = 90 degrees (since OQ is the radius and TQ is the tangent).

3. **Consider the Quadrilateral OPTQ:**
- The sum of the interior angles of a quadrilateral is 360 degrees. Thus, we can write:
\[ \text{Angle OPT} + \text{Angle POQ} + \text{Angle OQT} + \text{Angle PTQ} = 360^\circ \]

4. **Substitute Known Values:**
- We know:
- Angle OPT = 90 degrees
- Angle POQ = 110 degrees
- Angle OQT = 90 degrees
- Substituting these values into the equation gives:
\[ 90 + 110 + 90 + \text{Angle PTQ} = 360 \]

5. **Simplify the Equation:**
- Adding the known angles:
\[ 290 + \text{Angle PTQ} = 360 \]

6. **Solve for Angle PTQ:**
- To find Angle PTQ, we subtract 290 from both sides:
\[ \text{Angle PTQ} = 360 - 290 \]
\[ \text{Angle PTQ} = 70^\circ \]

### Final Answer:
Thus, the measure of angle PTQ is **70 degrees**.