In triangle ABC, an exterior angle at vertex C measures 127°. If angle A measures 88°, what is the measure,...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC has an exterior angle at vertex C measuring \(127°\)
  • Interior angle A measures \(88°\)
  • Need to find interior angle B

2. INFER the correct theorem to apply

• This is asking about an exterior angle and interior angles of a triangle
• The exterior angle theorem applies here: An exterior angle equals the sum of the two remote interior angles
• The exterior angle at C is "remote" from angles A and B (not adjacent to the exterior angle)

3. TRANSLATE the relationship into an equation

• Exterior angle at C = Angle A + Angle B
• \(127° = 88° + \mathrm{Angle\ B}\)

4. SIMPLIFY to solve for angle B

• Angle B = \(127° - 88° = 39°\)

Answer: A (39)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion: Students often confuse the exterior angle theorem with the triangle sum theorem (interior angles sum to \(180°\)).

Instead of using the exterior angle theorem, they might think: "The interior angles must sum to \(180°\), so if one exterior angle is \(127°\), then the interior angle at C is \(180° - 127° = 53°\). Then \(\mathrm{A} + \mathrm{B} + 53° = 180°\), so \(88° + \mathrm{B} + 53° = 180°\), which gives \(\mathrm{B} = 39°\)."

While this approach coincidentally leads to the correct answer in this case, it shows fundamental misunderstanding of the exterior angle theorem. This may lead them to select Choice A (39) for the wrong reasons, or in other problems, completely incorrect answers.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up \(127° = 88° + \mathrm{B}\) but make arithmetic errors.

Common calculation mistakes include \(127° - 88° = 41°\) or \(127° - 88° = 42°\), leading them to select Choice B (41) or Choice C (42).

The Bottom Line:

Success requires both recognizing this as an exterior angle theorem problem and executing the simple subtraction correctly. The conceptual leap to the exterior angle theorem is the key insight that unlocks the solution.