In the convex hexagon ABCDEF, side AB is parallel to side FE. The average of the measures of angle B...

1. TRANSLATE the problem information

• Given information:
  • Convex hexagon ABCDEF with AB || FE
  • Average of angles B and C is 131°
  • Angle E = 161°
  • Need to find angle D

• What this tells us:
  • Average of 131° means \(\frac{\angle \mathrm{B} + \angle \mathrm{C}}{2} = 131°\), so \(\angle \mathrm{B} + \angle \mathrm{C} = 262°\)
  • Parallel sides AB and FE create special angle relationships

2. INFER the approach

• Start with the fundamental polygon angle sum: For any hexagon, all interior angles sum to \((6-2) \times 180° = 720°\)

• Recognize the parallel line condition: When AB || FE with transversal AF, consecutive interior angles ∠A and ∠F are supplementary, giving us \(\angle \mathrm{A} + \angle \mathrm{F} = 180°\)

3. SIMPLIFY by organizing known angle relationships

• We now have:
  • \(\angle \mathrm{A} + \angle \mathrm{F} = 180°\) (from parallel lines)
  • \(\angle \mathrm{B} + \angle \mathrm{C} = 262°\) (from average condition)
  • \(\angle \mathrm{E} = 161°\) (given)
  • Total sum must equal 720°

4. SIMPLIFY the final calculation

• Set up the equation: \(\angle \mathrm{A} + \angle \mathrm{B} + \angle \mathrm{C} + \angle \mathrm{D} + \angle \mathrm{E} + \angle \mathrm{F} = 720°\)

• Group known sums: \((\angle \mathrm{A} + \angle \mathrm{F}) + (\angle \mathrm{B} + \angle \mathrm{C}) + \angle \mathrm{D} + \angle \mathrm{E} = 720°\)

• Substitute: \(180° + 262° + \angle \mathrm{D} + 161° = 720°\)

• Combine: \(603° + \angle \mathrm{D} = 720°\)

• Solve: \(\angle \mathrm{D} = 117°\)

Answer: B) 117

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students often misinterpret what 'AB is parallel to FE' means for angle relationships in a hexagon. They might think this creates equal corresponding angles rather than recognizing the supplementary consecutive interior angles formed by transversal AF.

Without this key relationship \(\angle \mathrm{A} + \angle \mathrm{F} = 180°\), they attempt to solve with insufficient information, often assuming angles A and F are equal (90° each) or trying to use other parallel line properties that don't apply here. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up all relationships but make arithmetic errors in the final steps. They might calculate \(180° + 262° + 161°\) incorrectly as 593° instead of 603°, leading to \(\angle \mathrm{D} = 127°\) instead of 117°. Since 127° isn't among the choices, this causes them to second-guess their entire approach and potentially select Choice C (125°) as the closest option.

The Bottom Line:

This problem tests whether students can connect parallel line properties to polygon angle sums. The key insight is recognizing that parallel sides in a polygon create supplementary consecutive interior angles, which provides the missing constraint needed to solve the system.