If a chord PQ subtends an angle of `80^@` at the centre of a circle, then angle between the tangents at P and Q is also `80^@`
State true or false

Answer

To determine whether the statement "If a chord PQ subtends an angle of 80° at the center of a circle, then the angle between the tangents at P and Q is also 80°" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circle and Chord**:
- Let O be the center of the circle, and let PQ be the chord that subtends an angle of 80° at the center O.

2. **Draw the Tangents**:
- Draw the tangents at points P and Q. Let M be an external point where the tangents meet.

3. **Identify Angles**:
- The angle subtended by the chord PQ at the center O is ∠POQ = 80°.
- The angles formed by the radius and the tangents are right angles (90°). Thus, ∠OPM = 90° and ∠OQM = 90°.

4. **Using the Quadrilateral Property**:
- Consider the quadrilateral OPMQ. The sum of the angles in a quadrilateral is 360°.
- Therefore, we can write:
\[ \angle POQ + \angle OPM + \angle OQM + \angle PMQ = 360° \]
- Substituting the known angles:
\[ 80° + 90° + 90° + \angle PMQ = 360° \]

5. **Calculate the Angle PMQ**:
- Simplifying the equation:
\[ 80° + 180° + \angle PMQ = 360° \]
\[ 260° + \angle PMQ = 360° \]
\[ \angle PMQ = 360° - 260° = 100° \]

6. **Conclusion**:
- The angle between the tangents at points P and Q (∠PMQ) is 100°, not 80°.
- Therefore, the statement is **false**.

### Final Answer:
The statement is **false**.