If S is the incentre of triangle ABC and ∠A = 30°, then the value of ∠BSC is:

Question

Accepted Answer

To determine the value of ∠BSC when S is the incentre of triangle ABC and ∠A = 30°, we need to use some properties of triangles and their incentres.

### Explanation:

1. **Incentre Properties**: The incentre S of a triangle is the point where the angle bisectors of the triangle intersect. It is equidistant from all sides of the triangle.

2. **Angle Sum Property**: In any triangle, the sum of the internal angles is always 180°. Therefore, in triangle ABC, we have:
   $$
   \angle A + \angle B + \angle C = 180^\circ
   $$
   Given that ∠A = 30°, we have:
   $$
   30^\circ + \angle B + \angle C = 180^\circ
   $$
   $$
   \angle B + \angle C = 150^\circ
   $$

3. **Angle ∠BSC Calculation**: The angle ∠BSC can be calculated using the formula for the angle formed by the incentre with two vertices of the triangle:
   $$
   \angle BSC = 90^\circ + \frac{\angle A}{2}
   $$
   Substituting the given value of ∠A:
   $$
   \angle BSC = 90^\circ + \frac{30^\circ}{2} = 90^\circ + 15^\circ = 105^\circ
   $$

Thus, the value of ∠BSC is 105°, which corresponds to Option 3.

### Why Other Options Are Incorrect:

- **Option 1 (90°)**: This option is incorrect because it does not account for the additional angle contributed by half of ∠A. The formula for ∠BSC includes the term $\frac{\angle A}{2}$, which adds 15° to 90°, resulting in 105°.

- **Option 2 (100°)**: This option is incorrect because it does not match the calculated value of 105°. The calculation clearly shows that the angle is 105°, not 100°.

- **Option 4 (110°)**: This option is incorrect because it overestimates the angle. The correct calculation shows that the angle is 105°, not 110°.

In conclusion, based on the properties of the incentre and the given angle ∠A, the correct value of ∠BSC is 105°, making Option 3 the correct choice.