To a circle with centre at 0, two tangents AP and AQ are drawn from an external point A. If the `angle`PAQ = `80^(@)`, then the `angle`POQ is:

Options

A. `80^(@)`
B. `70^(@)`
C. `90^(@)`
D. `100^(@)`

Answer

To solve the problem, we will follow these steps:

1. **Understanding the Geometry**: We have a circle with center O, and two tangents AP and AQ drawn from an external point A. The angle PAQ is given as 80 degrees.

2. **Identifying Right Angles**: Since AP and AQ are tangents to the circle, they meet the radius at points P and Q at right angles. Therefore, we have:
- Angle AOP = 90 degrees
- Angle AOQ = 90 degrees

3. **Setting Up the Angles**: The angles around point A can be expressed as:
- Angle PAQ + Angle AOP + Angle AOQ + Angle POQ = 360 degrees
- Substituting the known values, we have:
- 80 degrees + 90 degrees + 90 degrees + Angle POQ = 360 degrees

4. **Calculating Angle POQ**: Now, we can simplify the equation:
- 80 + 90 + 90 + Angle POQ = 360
- 260 + Angle POQ = 360
- Angle POQ = 360 - 260
- Angle POQ = 100 degrees

Thus, the angle POQ is 100 degrees.

### Summary of Steps:
1. Recognize the properties of tangents and the circle.
2. Identify the right angles formed by the tangents and the radius.
3. Set up the equation for the angles around point A.
4. Solve for the unknown angle POQ.