In triangle ABC, OB and OC are the internal bisectors of angles ∠B and ∠C respectively, meeting at point O. If angle ∠BAC = 60°, what is the measure (in degrees) of angle ∠BOC?

Introduction / Context:
This geometry question deals with angle bisectors inside a triangle and the special point where they intersect. That point is the incenter. The question uses a well known relationship between the angle at the incenter and the opposite vertex angle of the triangle.

Given Data / Assumptions:

• Triangle ABC is any triangle.
• OB is the internal bisector of angle ∠B.
• OC is the internal bisector of angle ∠C.
• These two bisectors meet at point O, the incenter.
• Angle ∠BAC = 60°.
• We must find angle ∠BOC.

Concept / Approach:
The incenter is the intersection point of the three internal angle bisectors of a triangle. A key property is that the angle between the bisectors from vertices B and C at the incenter equals 90° plus half of the third angle, that is ∠BOC = 90° + ∠A / 2 where ∠A is the angle at vertex A of the triangle.

Step-by-Step Solution:
Step 1: Recognize that OB and OC are internal angle bisectors, so their intersection O is the incenter of triangle ABC.Step 2: Recall the formula for the incenter angle between the bisectors from B and C: ∠BOC = 90° + ∠A / 2.Step 3: Here ∠A = ∠BAC = 60°.Step 4: Substitute into the formula: ∠BOC = 90° + 60° / 2.Step 5: Compute 60° / 2 = 30°, then 90° + 30° = 120°.Step 6: Therefore the required angle is ∠BOC = 120°.

Verification / Alternative check:
You can imagine a specific triangle, for example an equilateral triangle with all angles 60°, and check that the incenter equals the circumcenter. In that case, ∠BOC would subtend the same arc as ∠BAC and numerical calculations also confirm the value of 120° using central angle properties.

Why Other Options Are Wrong:
150°, 100° and 90° do not satisfy the formula 90° + ∠A / 2 for ∠A = 60°. Option 60° simply repeats the vertex angle and ignores the incenter relationship.

Common Pitfalls:
Common mistakes include thinking that ∠BOC equals ∠A, or applying the formula with the wrong vertex angle. Some students also confuse the incenter with the circumcenter and misapply central angle results. Remember that the special formula 90° + ∠A / 2 applies to the incenter angle.

Final Answer:
The measure of angle ∠BOC is 120°.