In quadrilateral PQRS, the measure of angle P is 95°, the measure of angle Q is 73°, and the measure...

1. TRANSLATE the problem information

• Given information:
  • Angle P = \(95°\)
  • Angle Q = \(73°\)
  • Angle R = \(108°\)
  • Need to find angle S in quadrilateral PQRS

• What this tells us: We have three of the four interior angles and need the fourth.

2. INFER the approach

• Since we're dealing with a quadrilateral, we can use the fact that all interior angles must sum to \(360°\)
• Strategy: Set up an equation with all four angles equaling \(360°\), then solve for the unknown

3. TRANSLATE to mathematical notation

• Write the angle sum equation: \(\mathrm{P + Q + R + S = 360°}\)
• Substitute known values: \(\mathrm{95° + 73° + 108° + S = 360°}\)

4. SIMPLIFY to find the answer

• Add the known angles: \(\mathrm{95° + 73° + 108° = 276°}\)
• Rewrite equation: \(\mathrm{276° + S = 360°}\)
• Solve for S: \(\mathrm{S = 360° - 276° = 84°}\)

Answer: B. 84°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when adding the three given angles.

For example, they might calculate \(\mathrm{95° + 73° + 108° = 274°}\) instead of \(\mathrm{276°}\), then solve \(\mathrm{S = 360° - 274° = 86°}\).

This may lead them to select Choice C (86°).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that quadrilaterals have interior angles summing to \(360°\).

Some students incorrectly think all polygons follow the triangle rule (\(180°\)), leading to confusion when they get a negative result: \(\mathrm{S = 180° - 276° = -96°}\). Since this doesn't make sense, this leads to confusion and guessing.

The Bottom Line:

This problem is straightforward once you know the quadrilateral angle sum property, but requires careful arithmetic execution to avoid simple calculation errors that can lead to plausible-looking wrong answers.