To draw a pair of tangents to a circle, which are inclined to each other at an angle of `45^(@)`, we have to draw tangents at the end points of those two radii, the angle between which is

Options

A. `105^(@)`
B. `135^(@)`
C. `140^(@)`
D. `145^(@)`

Explanation

To solve the problem of finding the angle between the two radii at the endpoints of the tangents that are inclined to each other at an angle of \(45^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Tangents and Radii**:
- We have a circle with center O.
- Let the points where the tangents touch the circle be A and B.
- The tangents at points A and B are inclined to each other at an angle of \(45^\circ\).

2. **Using the Property of Tangents**:
- The angle between the radius and the tangent at the point of contact is \(90^\circ\).
- Therefore, the angle \(OAP\) (between radius OA and tangent AP) is \(90^\circ\) and the angle \(OBP\) (between radius OB and tangent BP) is also \(90^\circ\).

3. **Forming a Cyclic Quadrilateral**:
- The points O, A, B, and P form a cyclic quadrilateral.
- In a cyclic quadrilateral, the sum of the opposite angles is \(180^\circ\).

4. **Setting Up the Angles**:
- Let angle AOB be the angle between the two radii OA and OB.
- The angles at points A and B are \(90^\circ\) each, and angle APB is given as \(45^\circ\).

5. **Applying the Cyclic Quadrilateral Property**:
- According to the property, we have:
\[ \text{Angle AOB} + \text{Angle APB} = 180^\circ \]
- Substituting the known angle:
\[ \text{Angle AOB} + 45^\circ = 180^\circ \]

6. **Solving for Angle AOB**:
- Rearranging the equation gives:
\[ \text{Angle AOB} = 180^\circ - 45^\circ = 135^\circ \]

### Final Answer:
The angle between the two radii OA and OB is \(135^\circ\).

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