In triangle ABC, side BC is produced to point D. If the exterior angle ∠ACD = 114° and angle ∠ABC is equal to half of angle ∠BAC, what is the measure of ∠BAC (in degrees)?

Introduction / Context:
This question combines the exterior angle property of triangles with a simple algebraic relation between interior angles. When one side of a triangle is extended, the angle formed outside the triangle is connected to the interior angle at that vertex. At the same time, we are told that angle ∠ABC is half of angle ∠BAC. By translating these relationships into equations, we can solve for the unknown angle ∠BAC using basic algebra.

Given Data / Assumptions:
- Triangle ABC is a general triangle. - Side BC is extended beyond C to a point D. - Exterior angle at C, ∠ACD, is 114°. - Interior angle at B, ∠ABC, is equal to (1 / 2) * ∠BAC. - We must find the measure of ∠BAC in degrees.

Concept / Approach:
The key geometric fact is that the exterior angle at a vertex on a straight line equals 180° minus the interior angle at that vertex. Thus angle ∠ACD and interior angle ∠ACB are supplementary. Therefore ∠ACB = 180° − ∠ACD. Once we know angle ∠ACB, we let ∠BAC be x, so ∠ABC is x / 2 by the given condition. The sum of the three interior angles must be 180°, leading to an equation in x. Solving that equation gives ∠BAC directly.

Step-by-Step Solution:
Step 1: Since BC is extended to D, angles ∠ACD and ∠ACB form a linear pair. Step 2: Therefore ∠ACB = 180° − ∠ACD = 180° − 114° = 66°. Step 3: Let ∠BAC = x degrees. Step 4: Given ∠ABC = (1 / 2) * ∠BAC, so ∠ABC = x / 2 degrees. Step 5: Use the angle sum property of a triangle: ∠BAC + ∠ABC + ∠ACB = 180°. Step 6: Substitute the expressions: x + (x / 2) + 66 = 180. Step 7: Combine like terms to get (3x / 2) + 66 = 180. Step 8: Subtract 66 from both sides: (3x / 2) = 114. Step 9: Multiply both sides by 2 / 3 to get x = (114 * 2) / 3 = 228 / 3 = 76.

Verification / Alternative check:
With x = 76°, we have ∠BAC = 76°, ∠ABC = 76 / 2 = 38° and ∠ACB = 66°. Check the angle sum: 76 + 38 + 66 = 180°, which satisfies the triangle angle sum property. Also, ∠ACD as an exterior angle equals 180° − 66° = 114°, which matches the given data. Both conditions are satisfied, so the computed value for ∠BAC is consistent with all constraints in the problem.

Why Other Options Are Wrong:
Option 36° would give ∠ABC = 18° and an angle sum that does not reach 180° once the 66° at C is included. Option 48° leads to ∠ABC = 24°, making the sum 48 + 24 + 66 = 138°, which is too small. Option 84° gives ∠ABC = 42°, and the sum 84 + 42 + 66 = 192°, which is larger than 180° and thus invalid.

Common Pitfalls:
Students sometimes confuse the exterior angle theorem and mistakenly add the interior and exterior angles instead of using their supplementary relationship. Another common error is to misinterpret the condition ∠ABC = (1 / 2) * ∠BAC and treat it in reverse. Forgetting the basic angle sum rule or mismanaging fractions in the algebra can also cause incorrect results. Writing down the equation carefully and solving step by step helps avoid these problems.

Final Answer:
The measure of angle ∠BAC is 76°.