In a circle, PQ and RS are two parallel tangents at A and B. The tangent at makes an intercept DE between PQ and RS. Then `angle DOE` is equal to : (where O is centre).

Options

A. `90^(@)`
B. `120^(@)`
C. `60^(@)`
D. `45^(@)`

Answer

To solve the problem, we need to find the measure of angle DOE, where D and E are points on the tangent DE that intercepts between the two parallel tangents PQ and RS, and O is the center of the circle.

### Step-by-step Solution:

1. **Identify the Elements**:
- We have a circle with center O.
- There are two parallel tangents PQ and RS at points A and B respectively.
- The tangent DE intersects between the two parallel tangents.

2. **Understanding Tangents**:
- The radius of the circle is perpendicular to the tangent at the point of tangency. Therefore, OA ⊥ PQ and OB ⊥ RS.

3. **Angles at the Center**:
- Since OA and OB are radii of the circle, we know that angle AOB is 90 degrees because the tangents are parallel and the radius meets the tangent at a right angle.

4. **Triangles Involved**:
- Consider triangles OAD and OBE. Both triangles share the common radius (OA and OB) and the segments AD and BE are tangents from points A and B to points D and E respectively.

5. **Congruent Triangles**:
- By the property of tangents from a point to a circle being equal, we have AD = AM and BE = BM.
- Therefore, triangles OAD and OBE are congruent by the Side-Angle-Side (SAS) criterion.

6. **Finding Angle DOE**:
- Since triangles OAD and OBE are congruent, angle OAD = angle OBE.
- Also, since OA and OB are both perpendicular to the tangents at points A and B, we have angle OAD = angle OBE = 90 degrees.

7. **Using Linear Pair**:
- In triangle DOE, we know that angle AOB = 90 degrees. Therefore, angle DOE must also be 90 degrees, as it is formed by the tangents DE intersecting at points D and E.

8. **Conclusion**:
- Thus, the measure of angle DOE is 90 degrees.

### Final Answer:
Angle DOE = 90 degrees.