In right triangle PQR, the right angle is at Q. If angleP measures 32°, what is the measure of angleR,...

1. TRANSLATE the problem information

• Given information:
  • Right triangle PQR with right angle at Q
  • \(\angle\mathrm{P} = 32°\)
  • Need to find \(\angle\mathrm{R}\)

• What this tells us:
  • \(\angle\mathrm{Q} = 90°\) (definition of right angle)
  • We have two known angles and need the third

2. INFER the approach

• Since we know two angles in a triangle and need the third, we should use the triangle angle sum property
• Strategy: Set up the equation \(\angle\mathrm{P} + \angle\mathrm{Q} + \angle\mathrm{R} = 180°\) and solve for \(\angle\mathrm{R}\)

3. SIMPLIFY using the triangle angle sum equation

• Set up: \(\angle\mathrm{P} + \angle\mathrm{Q} + \angle\mathrm{R} = 180°\)
• Substitute known values: \(32° + 90° + \angle\mathrm{R} = 180°\)
• Combine: \(122° + \angle\mathrm{R} = 180°\)
• Solve: \(\angle\mathrm{R} = 180° - 122° = 58°\)

Answer: 58

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misinterpret "the right angle is at Q" and either:

• Assume \(\angle\mathrm{Q} = 32°\) instead of \(\angle\mathrm{Q} = 90°\), or
• Set \(\angle\mathrm{P} = 90°\) instead of \(\angle\mathrm{Q} = 90°\)

This leads to setting up incorrect equations like \(90° + 32° + \angle\mathrm{R} = 180°\), giving \(\angle\mathrm{R} = 58°\) by coincidence, or other wrong values that cause confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that the sum of angles in any triangle is 180°, so they get stuck after correctly identifying the known angles. Without this fundamental relationship, they have no systematic way to find the third angle.

This causes them to abandon systematic solution and guess among reasonable angle measures.

The Bottom Line:

This problem tests whether students can connect basic triangle properties with right angle definitions. The key insight is recognizing that "right angle at Q" gives you \(\angle\mathrm{Q} = 90°\), then applying the triangle angle sum property becomes straightforward.