Two perpendicular lines intersect at point O, forming four 90° angles. A transversal line intersects each of the two perpendicular...

1. TRANSLATE the geometric setup into mathematical relationships

• Given information:
  • Two perpendicular lines create 4 right angles (\(90°\) each) at point O
  • A transversal creates 8 additional angles: 4 acute (all equal) and 4 obtuse (all equal)
  • One acute angle measures \((2\mathrm{x} + 30)°\)

• What this tells us:
  • We have 12 total angles: 4 right \(90°\), 4 acute \((2\mathrm{x} + 30)°\), and 4 obtuse
  • Since supplementary angles sum to \(180°\): each obtuse angle = \(180° - (2\mathrm{x} + 30)° = (150 - 2\mathrm{x})°\)

2. APPLY CONSTRAINTS to find the valid range for x

• Since the angles described as 'acute' must actually be acute:
  \(0 \lt 2\mathrm{x} + 30 \lt 90\)
  \(-30 \lt 2\mathrm{x} \lt 60\)
  \(-15 \lt \mathrm{x} \lt 30\)

3. CONSIDER ALL CASES by finding possible sums of any 4 angles

• INFER that we need to systematically check combinations:

Major combinations:

• 4 right angles: \(4(90) = 360°\)
• 4 acute angles: \(4(2\mathrm{x} + 30) = 8\mathrm{x} + 120°\)
• 4 obtuse angles: \(4(150 - 2\mathrm{x}) = 600 - 8\mathrm{x}°\)
• 2 right + 2 obtuse: \(2(90) + 2(150 - 2\mathrm{x}) = 180 + 300 - 4\mathrm{x} = 480 - 4\mathrm{x}°\)

Note: Many other combinations (like 2 right + 1 acute + 1 obtuse) also equal \(360°\)

4. SIMPLIFY and compare with answer choices

• Available expressions: \(360°\), \(8\mathrm{x} + 120°\), \(600 - 8\mathrm{x}°\), \(480 - 4\mathrm{x}°\), and others
• Checking choices:
  • (B) \(8\mathrm{x} + 120\) ✓ (4 acute angles)
  • (C) \(-8\mathrm{x} + 600\) ✓ (same as \(600 - 8\mathrm{x}\), which is 4 obtuse angles)
  • (D) \(360\) ✓ (multiple combinations)
  • (A) \(-4\mathrm{x} + 520\): The closest we can get with a \(-4\mathrm{x}\) term is \(480 - 4\mathrm{x}\), not \(520 - 4\mathrm{x}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often check only the most obvious combinations (like 4 of the same type of angle) and miss that many different combinations can yield the same sum.

They might see that choice (D) \(360°\) works for '4 right angles' and incorrectly conclude that since other choices also represent valid single-type combinations, all choices must be possible. They fail to systematically verify that choice (A) actually cannot be achieved by any combination.

This leads to confusion and guessing rather than definitively identifying the impossible expression.

The Bottom Line:

This problem requires methodical case analysis - you must systematically explore combinations rather than just checking whether each expression 'looks reasonable.' The key insight is that while several expressions can be formed multiple ways, choice (A) has no valid path to formation.