A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of these...

1. TRANSLATE the problem setup

• Given information:
  • A transversal intersects two parallel lines
  • Creates 4 acute angles and 4 obtuse angles (8 total)
  • One angle measures \((7\mathrm{x} - 250)°\)
  • Sum of 4 angles equals \(\mathrm{k}°\)
• What we need: Determine which expression could NOT equal k

2. INFER the angle relationships

• Key insight: When a transversal cuts parallel lines:
  • All 4 acute angles have the same measure
  • All 4 obtuse angles have the same measure
  • Each acute angle is supplementary to each obtuse angle
• Since one angle is \((7\mathrm{x} - 250)°\), its supplementary angle is:
  \(180 - (7\mathrm{x} - 250) = -7\mathrm{x} + 430°\)

3. CONSIDER ALL CASES for selecting 4 angles

• We need to find all possible ways to pick 4 angles from:
  • 4 angles of measure \((7\mathrm{x} - 250)°\)
  • 4 angles of measure \((-7\mathrm{x} + 430)°\)

The combinations are:

• Case 1: 4 acute angles
• Case 2: 3 acute + 1 obtuse
• Case 3: 2 acute + 2 obtuse
• Case 4: 1 acute + 3 obtuse
• Case 5: 4 obtuse angles

4. SIMPLIFY each case algebraically

• Case 1: \(4(7\mathrm{x} - 250) = 28\mathrm{x} - 1,000\)
• Case 2:
  \(3(7\mathrm{x} - 250) + (-7\mathrm{x} + 430)\)
  \(= 21\mathrm{x} - 750 - 7\mathrm{x} + 430\)
  \(= 14\mathrm{x} - 320\)
• Case 3:
  \(2(7\mathrm{x} - 250) + 2(-7\mathrm{x} + 430)\)
  \(= 14\mathrm{x} - 500 - 14\mathrm{x} + 860\)
  \(= 360\)
• Case 4:
  \((7\mathrm{x} - 250) + 3(-7\mathrm{x} + 430)\)
  \(= 7\mathrm{x} - 250 - 21\mathrm{x} + 1,290\)
  \(= -14\mathrm{x} + 1,040\)
• Case 5: \(4(-7\mathrm{x} + 430) = -28\mathrm{x} + 1,720\)

5. INFER which answer choice doesn't match

• Possible k values: \(28\mathrm{x} - 1,000\); \(14\mathrm{x} - 320\); \(360\); \(-14\mathrm{x} + 1,040\); \(-28\mathrm{x} + 1,720\)
• Comparing with choices:
  • B: \(14\mathrm{x} - 320\) ✓
  • C: \(-28\mathrm{x} + 1,720\) ✓
  • D: \(360\) ✓
  • A: \(-14\mathrm{x} + 1,540\) ✗ (should be \(-14\mathrm{x} + 1,040\))

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize that all acute angles are equal and all obtuse angles are equal when parallel lines are cut by a transversal. Instead, they might think each of the 8 angles could have different measures, making the problem much more complex than it actually is. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify the angle relationships but make algebraic errors when calculating the supplementary angle (getting \(7\mathrm{x} + 430\) instead of \(-7\mathrm{x} + 430\)) or when combining terms in the different cases. This computational error might lead them to select Choice B \((14\mathrm{x} - 320)\) if they miscalculate one of the valid cases.

The Bottom Line:

This problem tests understanding of parallel line properties combined with systematic case analysis. Success requires recognizing the geometric constraints that limit the possible angle measures, then carefully working through all combinations.