In triangle ABC, if angle A + angle C = 140° and angle A + 3 × angle B = 180°, then what is the measure of angle A (in degrees)?

Introduction / Context:
This algebraic geometry problem tests how to use angle relationships and the angle sum property of a triangle to solve for unknown angles. It involves forming and solving simple linear equations in terms of the triangle's angles.

Given Data / Assumptions:

• Triangle ABC has interior angles A, B, and C.
• Angle A + angle C = 140°.
• Angle A + 3 × angle B = 180°.
• We must find angle A.

Concept / Approach:
The key is to use the angle sum property A + B + C = 180° together with the given equations. From A + C = 140°, we can express B in terms of given values. This allows us to find B first and then substitute back into the second equation A + 3B = 180° to find A. The approach consists of solving two linear equations with two unknowns (A and B) using substitution.

Step-by-Step Solution:
Step 1: Use the triangle angle sum property: A + B + C = 180°. Step 2: From A + C = 140°, subtract from the sum: (A + B + C) − (A + C) = 180° − 140°, which gives B = 40°. Step 3: Now use the second given relation: A + 3B = 180°. Step 4: Substitute B = 40°: A + 3 × 40° = 180°. Step 5: Simplify: A + 120° = 180°. Step 6: Subtract 120° from both sides: A = 60°.

Verification / Alternative Check:
With A = 60° and B = 40°, we can find C from A + C = 140°: 60° + C = 140°, so C = 80°. Check the angle sum: A + B + C = 60° + 40° + 80° = 180°, which is correct. Also check A + 3B: 60° + 3 × 40° = 60° + 120° = 180°, matching the second condition. All constraints are satisfied, so A = 60° is consistent.

Why Other Options Are Wrong:
If A were 80°, then from A + C = 140° we would get C = 60°, and from A + 3B = 180° we would get 3B = 100° so B would not be an integer and the angle sum would fail. Similar contradictions arise for A = 40° or A = 20°, where one or both of the given relationships would not hold once the remaining angles are computed. Only A = 60° satisfies all equations at once.

Common Pitfalls:
A frequent mistake is to try guessing angle values without systematically using the angle sum property and both given equations. Another pitfall is mismanaging algebra when subtracting or dividing, which can lead to incorrect values for B or C. Writing out each step carefully and checking all given conditions after finding an answer helps avoid such errors.

Final Answer:
The measure of angle A is 60°.