In quadrilateral PQRS, the measure of angleP is 58°, the measure of angleQ is 87°, and the measure of angleR...

1. TRANSLATE the problem information

• Given information:
  • Quadrilateral PQRS with three known angles
  • \(\angle\mathrm{P} = 58°\), \(\angle\mathrm{Q} = 87°\), \(\angle\mathrm{R} = 94°\)
  • Need to find \(\angle\mathrm{S}\)

2. INFER the approach needed

• Since we have a quadrilateral with three known angles, we can use the angle sum property
• The sum of all interior angles in any quadrilateral equals 360°
• We can set up an equation to solve for the missing angle

3. TRANSLATE this insight into an equation

• Write: \(\angle\mathrm{P} + \angle\mathrm{Q} + \angle\mathrm{R} + \angle\mathrm{S} = 360°\)
• Substitute known values: \(58° + 87° + 94° + \angle\mathrm{S} = 360°\)

4. SIMPLIFY through arithmetic

• First, add the known angles: \(58° + 87° + 94° = 239°\)
• The equation becomes: \(239° + \angle\mathrm{S} = 360°\)
• Solve for \(\angle\mathrm{S}\): \(\angle\mathrm{S} = 360° - 239° = 121°\)

Answer: D. 121°

Why Students Usually Falter on This Problem

Most Common Error Path:

Arithmetic errors in SIMPLIFY: Students correctly set up the equation but make calculation mistakes when adding the three given angles.

For example, they might calculate \(58° + 87° + 94°\) as 249° instead of 239°, leading to \(\angle\mathrm{S} = 360° - 249° = 111°\). This leads them to select Choice B (111°).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that the sum of interior angles in a quadrilateral is 360°.

Without this key formula, they cannot set up the problem correctly and may attempt to use incorrect relationships or guess randomly among the answer choices.

The Bottom Line:

This problem tests both conceptual knowledge (quadrilateral angle sum) and careful arithmetic execution. Success requires knowing the fundamental property and avoiding computational errors in multi-step addition and subtraction.