In triangle ABC, side BC is produced to a point D so that CD lies on the extension of BC. If the exterior angle ∠ACD = 120° and angle ∠B is equal to half of angle ∠A (that is, ∠B = (1/2)∠A), then what is the measure of angle ∠A (in degrees)?

Introduction / Context:
This question combines the exterior angle property of triangles with a simple algebraic relationship between two interior angles. In many aptitude and geometry problems, you are given one exterior angle and some relationship between the interior angles, and you must apply the basic angle sum and exterior angle rules to find the required vertex angle. Understanding that an exterior angle of a triangle equals the sum of the two opposite interior angles is crucial for solving such problems efficiently.

Given Data / Assumptions:
• Triangle ABC is given, and side BC is extended to point D to form an exterior angle at C.
• The exterior angle at C is ∠ACD = 120°.
• Angle at vertex B satisfies ∠B = (1 / 2) ∠A, where ∠A is ∠BAC.
• We need to determine the measure of angle ∠A in degrees.
• The triangle is assumed to be a standard Euclidean triangle with positive angles summing to 180°.

Concept / Approach:
The main geometric fact used here is the exterior angle theorem: in any triangle, an exterior angle equals the sum of the two interior opposite angles. When BC is extended to D, ∠ACD is the exterior angle at C, and its two remote interior angles are ∠A and ∠B. Therefore, we have ∠ACD = ∠A + ∠B. We are also given a direct algebraic relationship between ∠A and ∠B, namely ∠B = (1 / 2) ∠A. Substituting this relationship into the exterior angle equation produces a simple linear equation in ∠A, which we can solve to get the required angle.

Step-by-Step Solution:
Step 1: Use the exterior angle property for triangle ABC with BC produced to D: ∠ACD = ∠A + ∠B. Step 2: Substitute the given values into the relationship: 120° = ∠A + ∠B. Step 3: Use the given condition ∠B = (1 / 2) ∠A. Step 4: Replace ∠B in the equation: 120° = ∠A + (1 / 2) ∠A. Step 5: Combine like terms: 120° = (3 / 2) ∠A. Step 6: Solve for ∠A: ∠A = (2 / 3) * 120° = 80°.

Verification / Alternative check:
Once we find ∠A = 80°, we can calculate ∠B from the given relationship ∠B = (1 / 2) ∠A, so ∠B = 40°. The exterior angle at C should equal the sum of these two interior angles, so ∠ACD = ∠A + ∠B = 80° + 40° = 120°, which matches the given value. Furthermore, the interior angles of triangle ABC should sum to 180°, so ∠C = 180° - (80° + 40°) = 60°, which is a valid positive angle. Everything is consistent, so ∠A = 80° is confirmed.

Why Other Options Are Wrong:
If ∠A were 60°, then ∠B would be 30°, and their sum would be 90°, not 120°. If ∠A were 75°, then ∠B would be 37.5°, and their sum would be 112.5°, again not 120°. If ∠A were 90°, then ∠B would be 45°, and the sum would be 135°. None of these combinations satisfy the exterior angle equation ∠ACD = 120°. Only ∠A = 80° leads to ∠A + ∠B = 120° and a valid triangle angle set.

Common Pitfalls:
One frequent mistake is to assume that the exterior angle is directly supplementary to one interior angle only, and then attempt to solve using 180° - 120° = 60° without using the given relationship with the second interior angle. Another error is misinterpreting the statement ∠B = (1 / 2) ∠A, either by inverting it or by using a wrong factor in the equation. To avoid these pitfalls, always write down the correct exterior angle relation and then carefully substitute the algebraic relationship between angles. Checking the sum of all interior angles at the end is a good way to diagnose mistakes.

Final Answer:
The measure of angle ∠A is 80°.