In triangle ABC, angle A measures 40° and can be expressed as \((2\mathrm{x} - 80)°\), while angle B measures \((\mathrm{x}...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Angle\;A = 40° = (2x - 80)°}\)
  • \(\mathrm{Angle\;B = (x + 20)°}\)
  • \(\mathrm{Exterior\;angle\;at\;C = (-x + w)°}\)
  • Need to find: \(\mathrm{w}\)

2. INFER that we need to find x first

• Since angle A has two expressions for the same measure, we can set them equal
• Once we have x, we can find the actual measures of the angles
• Then we can use the relationship between interior and exterior angles

3. SIMPLIFY to solve for x

• Set up the equation: \(\mathrm{2x - 80 = 40}\)
• Add 80 to both sides: \(\mathrm{2x = 120}\)
• Divide by 2: \(\mathrm{x = 60}\)

4. INFER the measures of the interior angles

• \(\mathrm{Angle\;A = 40°}\) (given)
• \(\mathrm{Angle\;B = x + 20 = 60 + 20 = 80°}\)

5. INFER that the exterior angle theorem applies

• The exterior angle at vertex C equals the sum of the two non-adjacent interior angles
• \(\mathrm{Exterior\;angle\;at\;C = Angle\;A + Angle\;B = 40° + 80° = 120°}\)

6. SIMPLIFY to find w

• The exterior angle at C also equals \(\mathrm{(-x + w)°}\)
• Set up equation: \(\mathrm{-x + w = 120}\)
• Substitute \(\mathrm{x = 60}\): \(\mathrm{-60 + w = 120}\)
• Add 60 to both sides: \(\mathrm{w = 180}\)

Answer: 180

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge of the exterior angle theorem: Students may not remember or apply the relationship that an exterior angle equals the sum of the two non-adjacent interior angles. Instead, they might try to use the fact that interior angles sum to 180°, which doesn't directly help with the exterior angle expression.

This leads to confusion and guessing, or attempting to use \(\mathrm{angle\;C = 180° - 40° - 80° = 60°}\) and incorrectly relating this to the exterior angle.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify that they need to solve \(\mathrm{2x - 80 = 40}\) for x, but make algebraic errors in the process, such as getting \(\mathrm{x = -20}\) instead of \(\mathrm{x = 60}\). This cascades through the rest of the problem, leading to an incorrect value for w.

This may lead them to select an incorrect answer or abandon the systematic approach and guess.

The Bottom Line:

This problem requires students to work systematically through multiple steps while keeping track of which angle expressions represent the same angle. The key insight is recognizing that the exterior angle theorem provides the crucial connection between the interior angles and the exterior angle expression.