PQ and RS are two parallel tangents to a circle with centre O and another tangent AB wih point of contact C intersect PQ at A and RS at B. then find `angle AOB`

Options

A. `60^(@)`
B. `120^(@)`
C. `90^(@)`
D. `180^(@)`

Answer

To solve the problem, we need to find the angle AOB formed by the tangents AB at points A and B, where PQ and RS are parallel tangents to a circle with center O.

### Step-by-Step Solution:

1. **Draw the Diagram**:
- Start by drawing a circle with center O.
- Draw two parallel tangents PQ and RS to the circle.
- Mark the points where the tangents touch the circle as C (the point of contact for tangent AB).
- Extend the tangent AB such that it intersects PQ at point A and RS at point B.

2. **Identify Angles**:
- Since PQ and RS are tangents to the circle, the radius OC (from the center O to the point of contact C) is perpendicular to the tangent AB at point C. Therefore, angle OCA = 90° and angle OCB = 90°.

3. **Use Properties of Tangents**:
- The angles formed by the tangents at points A and B with the line segments OA and OB (i.e., angle OAC and angle OBC) are equal because they are angles made by the same external point A and B with the tangents. Let's denote angle OAC as angle 1 and angle OBC as angle 2.
- Thus, angle 1 = angle 2.

4. **Consider the Quadrilateral**:
- The angles in quadrilateral formed by points A, B, C, and the center O can be expressed as:
- Angle AOB + angle OAC + angle OBC + angle AOB = 360°.
- Since angle OAC and angle OBC are equal, we can denote them both as angle 1.
- Therefore, the equation simplifies to:
- Angle AOB + 90° + angle 1 + 90° = 360°.
- This means:
- Angle AOB + 180° + 2(angle 1) = 360°.
- Rearranging gives:
- Angle AOB + 2(angle 1) = 180°.

5. **Finding Angle AOB**:
- Since angle 1 = angle 2, we can substitute angle 2 for angle 1 in the equation:
- Angle AOB + 2(angle 1) = 180°.
- This implies:
- Angle AOB = 180° - 2(angle 1).
- Since angle 1 + angle 2 = 90° (from the properties of angles in a right triangle), we can conclude:
- Angle AOB = 90°.

### Conclusion:
Thus, the angle AOB is 90°.