In quadrilateral PQRS, the measure of angle P is 85°, the measure of angle Q is 95°, and the measure...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Angle\;P = 85°}\)
  • \(\mathrm{Angle\;Q = 95°}\)
  • \(\mathrm{Angle\;R = 70°}\)
  • \(\mathrm{Angle\;S = (2n + 10)°}\)
  • Need to find the value of n

2. INFER the key relationship

• Since PQRS is a quadrilateral, the sum of all interior angles must equal \(\mathrm{360°}\)
• This gives us the equation we need to solve for n

3. TRANSLATE into equation form

• Set up: \(\mathrm{P + Q + R + S = 360°}\)
• Substitute: \(\mathrm{85° + 95° + 70° + (2n + 10)° = 360°}\)

4. SIMPLIFY through algebraic steps

• First, add the known angles: \(\mathrm{85° + 95° + 70° = 250°}\)
• Equation becomes: \(\mathrm{250° + (2n + 10)° = 360°}\)
• Expand the parentheses: \(\mathrm{250° + 2n + 10° = 360°}\)
• Combine like terms: \(\mathrm{260° + 2n = 360°}\)
• Solve for the variable: \(\mathrm{2n = 360° - 260° = 100°}\)
• Therefore: \(\mathrm{n = 50}\)

Answer: A (50)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not immediately recognize that they need to use the sum of interior angles property for quadrilaterals. Some students might try to find individual angle relationships or get overwhelmed by the algebraic expression \(\mathrm{(2n + 10)°}\), leading them to abandon a systematic approach and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equation but make arithmetic errors. Common mistakes include:

• Incorrectly adding \(\mathrm{85° + 95° + 70°}\) (getting something other than \(\mathrm{250°}\))
• Making sign errors when moving terms: writing \(\mathrm{2n = 260° - 360°}\) instead of \(\mathrm{2n = 360° - 260°}\)
• Forgetting to add the \(\mathrm{10°}\) when combining \(\mathrm{250° + 10° = 260°}\)

These calculation errors typically lead to wrong values for n, causing them to select incorrect answer choices.

The Bottom Line:

This problem tests whether students can connect the fundamental property of quadrilaterals (angle sum = \(\mathrm{360°}\)) with basic algebraic manipulation. Success depends on both recognizing the right mathematical relationship and executing the algebra carefully.