In an isosceles triangle ABC, sides AB and AC are equal and ∠A = 80°. The internal bisectors of angles ∠B and ∠C meet at point D inside the triangle. What is the measure (in degrees) of angle ∠BDC?

Introduction / Context:
In this geometry question we work with an isosceles triangle and angle bisectors meeting at a special interior point. The goal is to find the angle formed at that point between the two bisectors, which is a standard property related to the incenter of a triangle.

Given Data / Assumptions:

• Triangle ABC is isosceles with AB = AC.
• Angle ∠A = 80°.
• The internal bisectors of ∠B and ∠C meet at point D.
• We have to find the measure of angle ∠BDC.

Concept / Approach:
The intersection of the internal angle bisectors of a triangle is called the incenter. A key property is that the angle between the bisectors of ∠B and ∠C at the incenter is 90° + ∠A / 2. We use the fact that in an isosceles triangle with AB = AC, angles at B and C are equal.

Step-by-Step Solution:
Step 1: Since AB = AC, triangle ABC is isosceles with vertex at A, so ∠B = ∠C.Step 2: Sum of angles in a triangle is 180°, so ∠B + ∠C + ∠A = 180°.Step 3: With ∠A = 80° and ∠B = ∠C, we get 2∠B + 80° = 180°, so 2∠B = 100° and ∠B = ∠C = 50°.Step 4: The point where the internal bisectors of ∠B and ∠C meet is the incenter, which we call D.Step 5: A known result for the incenter is ∠BDC = 90° + ∠A / 2.Step 6: Substitute ∠A = 80° to get ∠BDC = 90° + 80° / 2 = 90° + 40° = 130°.

Verification / Alternative check:
Another way is to remember the general formulas: ∠BIC (incenter at I) equals 90° + ∠A / 2, and similarly for other vertex combinations. Substituting the given angle confirms that 130° is consistent with this standard incenter property.

Why Other Options Are Wrong:
90°: This would be true only if ∠A were 0°, which is impossible in a real triangle. 100° and 80° do not follow from the formula 90° + ∠A / 2. 60° is too small and does not match any central or interior angle relationship here.

Common Pitfalls:
A common mistake is to assume that ∠BDC equals either ∠B or ∠C or to average them. Another error is to forget that the meeting point of angle bisectors is the incenter and to ignore the special formula 90° + ∠A / 2. Some learners also miscompute half of 80° or misapply the triangle angle sum rule.

Final Answer:
The angle between the bisectors of ∠B and ∠C at point D is 130°.