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

1. INFER the angle relationships

When a line intersects two parallel lines (called a transversal), it creates 8 angles with special properties:

• All 4 acute angles are congruent (equal measures)
• All 4 obtuse angles are congruent (equal measures)
• Any acute angle and any obtuse angle are supplementary (sum to 180°)

2. TRANSLATE the given information

• Given information:
  • One acute angle measures \((9\mathrm{x} - 560)°\)
  • Sum of one acute angle + three obtuse angles = \((-18\mathrm{x} + \mathrm{w})°\)
  • Need to find: w

3. INFER the obtuse angle measure

Since acute and obtuse angles are supplementary:

• Obtuse angle = \(180° - \text{(acute angle)}\)
• Obtuse angle = \(180° - (9\mathrm{x} - 560)°\)
• Obtuse angle = \(180° - 9\mathrm{x} + 560° = (-9\mathrm{x} + 740)°\)

4. TRANSLATE the sum condition into an equation

Sum = one acute angle + three obtuse angles

• \((9\mathrm{x} - 560) + 3(-9\mathrm{x} + 740) = (-18\mathrm{x} + \mathrm{w})\)

5. SIMPLIFY to solve for w

Expand the left side:

• \((9\mathrm{x} - 560) + 3(-9\mathrm{x} + 740)\)
• \(= 9\mathrm{x} - 560 - 27\mathrm{x} + 2220\)
• \(= -18\mathrm{x} + 1660\)

Set equal to the right side:

• \(-18\mathrm{x} + 1660 = -18\mathrm{x} + \mathrm{w}\)
• Therefore: \(\mathrm{w} = 1660\)

Answer: 1660

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that all acute angles are equal and all obtuse angles are equal when parallel lines are cut by a transversal.

Students might think each of the 8 angles has a different measure, leading them to create unnecessarily complex equations with multiple variables. They may try to set up equations like "angle 1 + angle 2 + angle 3 + angle 4 = (-18x + w)" without understanding the congruent relationships. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Making algebraic errors when expanding \(3(-9\mathrm{x} + 740)\).

Students correctly identify the relationships but make errors like:

• Forgetting to distribute the 3 to both terms: getting \(3(-9\mathrm{x}) + 740\) instead of \(3(-9\mathrm{x} + 740)\)
• Sign errors when combining terms: getting \(-18\mathrm{x} - 1660\) instead of \(-18\mathrm{x} + 1660\)

This may lead them to calculate incorrect values for w.

The Bottom Line:

This problem tests whether students understand the geometric relationships created by parallel lines and transversals, then can translate those relationships into accurate algebraic expressions. The key insight is recognizing the pattern of congruent angles rather than treating each angle as independent.