If angle between two radii of a circle is `80^@` , what will be the angle . between the tangents at the ends of the radii ?

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have a circle with center O.
- Two radii OA and OB are drawn, forming an angle of 80 degrees at the center (∠AOB = 80°).

2. **Understanding the Geometry**:
- When tangents are drawn from points A and B to the circle, they will meet at an external point P.
- The angles formed between the tangents and the radii can be analyzed.

3. **Using Properties of Tangents**:
- The angle between the two tangents (APB) can be found by using the property that the angle between the tangents from an external point is equal to the sum of the angles subtended by the radii at the center.
- Therefore, we need to find the angles AOP and BOP.

4. **Finding Angles AOP and BOP**:
- Since OA and OB are equal (both are radii), triangles AOP and BOP are congruent.
- Given that ∠AOB = 80°, we can divide this angle equally between the two angles AOP and BOP.
- Thus, ∠AOP = ∠BOP = 80° / 2 = 40°.

5. **Finding Angles at Point P**:
- The angle between the radius OA and the tangent AP is 90° (since the radius is perpendicular to the tangent at the point of contact).
- Therefore, ∠PAO = 90°.

6. **Using Triangle Properties**:
- In triangle AOP, the sum of angles is 180°.
- We have:
- ∠AOP = 40°
- ∠PAO = 90°
- Let ∠APO = x.
- Therefore, we can write the equation:
- 40° + 90° + x = 180°
- 130° + x = 180°
- x = 180° - 130° = 50°.
- So, ∠APO = 50°.

7. **Finding the Angle APB**:
- Since ∠APO = 50° and ∠BPO (which is equal to ∠APO due to symmetry) is also 50°,
- The angle between the tangents at points A and B is:
- ∠APB = ∠APO + ∠BPO = 50° + 50° = 100°.

### Final Answer:
The angle between the tangents at the ends of the radii is **100 degrees**.

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