In $ riangle$ ΑВС, ∠A = 66°, the internal bisectors of ∠B and ∠C intersect at O. The measure of ∠BOC is :

Question

In $	riangle$ ΑВС, ∠A = 66°, the internal bisectors of ∠B and ∠C intersect at O. The measure of ∠BOC is :

Accepted Answer

To determine the measure of ∠BOC in triangle ABC, where ∠A = 66° and the internal bisectors of ∠B and ∠C intersect at O, we can use the properties of angle bisectors and the triangle's internal angles.

### Explanation:

1. **Sum of Angles in a Triangle:**
   In any triangle, the sum of the internal angles is always 180°. Therefore, in triangle ABC:
   $$
   ∠A + ∠B + ∠C = 180°
   $$
   Given that ∠A = 66°, we have:
   $$
   66° + ∠B + ∠C = 180°
   $$
   $$
   ∠B + ∠C = 114°
   $$

2. **Angle Bisector Theorem:**
   The internal bisectors of angles ∠B and ∠C intersect at point O. According to the angle bisector theorem, the measure of angle ∠BOC is given by:
   $$
   ∠BOC = 90° + \frac{∠A}{2}
   $$
   Substituting the value of ∠A:
   $$
   ∠BOC = 90° + \frac{66°}{2} = 90° + 33° = 123°
   $$

Thus, the measure of ∠BOC is 123°, which corresponds to Option 3.

### Why Other Options are Incorrect:

1. **Option 1: 57°**
   - This option is incorrect because it does not align with the calculation using the angle bisector theorem. The calculated angle ∠BOC is 123°, not 57°.

2. **Option 2: 114°**
   - This option is incorrect because it represents the sum of ∠B and ∠C, not the measure of ∠BOC. The angle ∠BOC is calculated using the angle bisector theorem and is 123°.

3. **Option 4: 130°**
   - This option is incorrect because it exceeds the calculated value of ∠BOC. The correct calculation using the angle bisector theorem gives 123°, not 130°.

In conclusion, the correct measure of ∠BOC is 123°, which is why Option 3 is the correct answer.