Question:Two lines intersect at a point, creating exactly two acute angles and two obtuse angles.The measure of one of the...

1. TRANSLATE the problem information

• Given information:
  • One acute angle measures \((8\mathrm{p} - 45)\) degrees
  • Sum of two obtuse angles and one acute angle is \((-8\mathrm{p} + \mathrm{w})\) degrees
  • Two lines intersect creating exactly 2 acute and 2 obtuse angles

2. INFER the geometric relationships

• When two lines intersect, they create four angles with special properties:
  • Vertical angles (opposite each other) are always equal
  • Adjacent angles are supplementary (add to \(180°\))
• This means we have two equal acute angles and two equal obtuse angles

3. INFER what we can determine about all angles

• Since both acute angles are equal: both measure \((8\mathrm{p} - 45)\) degrees
• Since adjacent angles are supplementary:
  \(\mathrm{Acute\ angle + Obtuse\ angle = 180°}\)
  \((8\mathrm{p} - 45) + \mathrm{Obtuse\ angle} = 180°\)
• So each obtuse angle \(= 180° - (8\mathrm{p} - 45) = 225 - 8\mathrm{p}\) degrees

4. SIMPLIFY to find the sum

• Sum of two obtuse angles + one acute angle:
  \(2(225 - 8\mathrm{p}) + (8\mathrm{p} - 45)\)
  \(= 450 - 16\mathrm{p} + 8\mathrm{p} - 45\)
  \(= 405 - 8\mathrm{p}\)

5. INFER the final step

• The sum equals \((-8\mathrm{p} + \mathrm{w})\), so:
  \(405 - 8\mathrm{p} = -8\mathrm{p} + \mathrm{w}\)
  Therefore: \(\mathrm{w} = 405\)

Answer: 405

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that intersecting lines create vertical angle pairs and supplementary adjacent angles.

Students might think all four angles are different or not understand the equal/supplementary relationships. Without these key geometric insights, they cannot set up the correct equations and end up guessing or using incorrect relationships.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when expanding \(2(225 - 8\mathrm{p}) + (8\mathrm{p} - 45)\).

Students might incorrectly distribute, combine like terms wrongly, or make sign errors. For example, getting \(450 - 16\mathrm{p} + 8\mathrm{p} + 45 = 495 - 8\mathrm{p}\) instead of \(405 - 8\mathrm{p}\), leading them to conclude \(\mathrm{w} = 495\).

The Bottom Line:

This problem tests whether students can connect geometric relationships to algebraic expressions. The key insight is recognizing how intersecting lines create predictable angle patterns, then translating those patterns into solvable equations.