In triangle ABC, an exterior angle at vertex B measures 142°. The measure of ∠A is 67°. What is the...

1. TRANSLATE the problem information

• Given information:
  • Exterior angle at vertex B = \(142°\)
  • \(\mathrm{∠A} = 67°\)
  • Need to find: \(\mathrm{∠C}\)

• What this tells us: We have one exterior angle and one interior angle, need to find another interior angle

2. INFER the appropriate relationship

• The key insight: Since we have an exterior angle and need to relate it to interior angles, we should use the exterior angle theorem
• The exterior angle theorem states: An exterior angle equals the sum of the two non-adjacent interior angles
• For our triangle: Exterior angle at B = \(\mathrm{∠A} + \mathrm{∠C}\)

3. TRANSLATE the theorem into an equation

Set up the equation:

\(142° = 67° + \mathrm{∠C}\)

4. SIMPLIFY to find the answer

Solve for \(\mathrm{∠C}\):

\(\mathrm{∠C} = 142° - 67°\)

\(\mathrm{∠C} = 75°\)

Answer: B. 75°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the exterior angle theorem with the interior angle sum theorem (interior angles of a triangle sum to \(180°\)).

They might think: "I know \(\mathrm{∠A} = 67°\), and all angles in a triangle add to \(180°\), so \(\mathrm{∠B} + \mathrm{∠C} = 180° - 67° = 113°\)." But since they don't know \(\mathrm{∠B}\), they might incorrectly assume \(\mathrm{∠C} = 113°\) by itself.

This may lead them to select Choice D (113°).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "exterior angle at vertex B" means and confuse it with the interior angle \(\mathrm{∠B}\).

If they think the interior \(\mathrm{∠B} = 142°\), they might use the interior angle sum: \(\mathrm{∠A} + \mathrm{∠B} + \mathrm{∠C} = 180°\), giving them \(67° + 142° + \mathrm{∠C} = 180°\), so \(\mathrm{∠C} = 180° - 209° = -29°\). Since this is impossible, they get confused and guess.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

Success on this problem requires clearly distinguishing between interior and exterior angles, and knowing which theorem applies to exterior angles. The exterior angle theorem is often less familiar to students than the interior angle sum, making theorem selection the key challenge.