Triangle ABC is isosceles with AB = AC and angle A = 40 degrees. Side BC is produced at both ends to points D and E, forming exterior angles ∠ABD and ∠ACE. The bisectors of these exterior angles meet at point O. What is the measure of angle BOC?

Introduction / Context:
This problem involves an isosceles triangle, exterior angles and angle bisectors intersecting at a point outside the triangle. It is more challenging than direct angle sum questions because it combines several geometry ideas. Solving it requires careful analysis of interior and exterior angle relationships and an understanding of symmetry in isosceles triangles.

Given Data / Assumptions:
- Triangle ABC is isosceles with AB = AC.
- Angle A is 40 degrees.
- Side BC is extended beyond B to D and beyond C to E.
- Exterior angles at B and C are ∠ABD and ∠ACE respectively.
- The bisectors of these exterior angles meet at point O.
- We need to find the measure of angle BOC at the intersection of these bisectors.

Concept / Approach:
In an isosceles triangle with AB = AC, the base angles at B and C are equal. The interior angle sum gives the base angles when angle A is known. Each exterior angle is supplementary to its corresponding interior angle. The bisectors of the exterior angles at B and C form equal angles with each side involved. Due to symmetry of the isosceles triangle, point O lies on the perpendicular bisector of BC, which simplifies the angle at O. We can use these facts and some geometric reasoning to find angle BOC.

Step-by-Step Solution:
Step 1: In triangle ABC, sum of angles is 180 degrees, so A + B + C = 180 degrees. Step 2: Given AB = AC, triangle ABC is isosceles with equal base angles B and C. Step 3: Angle A is 40 degrees, so B + C = 180 - 40 = 140 degrees. Therefore B = C = 70 degrees. Step 4: Exterior angle ∠ABD at B is supplementary to interior angle B, so ∠ABD = 180 - 70 = 110 degrees. Similarly, exterior angle ∠ACE at C is also 110 degrees. Step 5: The bisectors of these 110 degree exterior angles divide each into two angles of 55 degrees. So each bisector makes a 55 degree angle with AB or AC and a 55 degree angle with BC extended. Step 6: Because the triangle is symmetric about the axis through A and the midpoint of BC, the two bisectors are symmetric and intersect on this axis. When you compute the angle at their intersection BOC using geometry or a coordinate approach, you obtain ∠BOC = 70 degrees.

Verification / Alternative check:
An alternative way is to use a coordinate geometry construction. Place B and C symmetrically on the x axis and A above the axis so that AB = AC and angle A is 40 degrees. Compute the equations of the exterior angle bisectors at B and C and find their intersection point O. Then find angle BOC using vector dot product and inverse cosine. This method also yields 70 degrees, confirming the exact value found by geometric reasoning.

Why Other Options Are Wrong:
Angles 80, 100, 110 and 55 degrees do not satisfy the symmetric configuration and the way the exterior bisectors intersect. For example, 110 degrees is the size of the exterior angles themselves before bisection, not the angle at O. 55 degrees appears inside the construction but is not the final central angle BOC.

Common Pitfalls:
Learners often confuse interior and exterior angles or forget that the bisector divides the exterior angle, not the interior one. Another mistake is to assume that the angle at the intersection of bisectors is directly equal to one of the base angles without proper reasoning. Drawing a clear symmetric diagram and systematically tracking each angle helps avoid these errors.

Final Answer:
The measure of angle BOC is 70 degrees.