In triangle PQR, an exterior angle at vertex R measures 135°. The measure of angle P is 40°. What is...

1. TRANSLATE the problem information

• Given information:
  • Triangle PQR has an exterior angle at vertex R measuring \(135°\)
  • Interior angle P = \(40°\)
  • Need to find interior angle Q

2. INFER the most efficient approach

• We can use the exterior angle theorem directly: An exterior angle equals the sum of the two remote interior angles
• The exterior angle at R is "remote" from angles P and Q, so it equals P + Q
• This gives us a direct equation to solve

3. TRANSLATE the theorem into an equation

• Exterior angle at R = \(\angle\mathrm{P} + \angle\mathrm{Q}\)
• \(135° = 40° + \angle\mathrm{Q}\)

4. SIMPLIFY to find the answer

• \(\angle\mathrm{Q} = 135° - 40°\)
• \(\angle\mathrm{Q} = 95°\)

Answer: D (95°)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse interior and exterior angles, thinking the \(135°\) angle IS the interior angle at R.

If they think angle R = \(135°\), they would use the triangle sum theorem:

\(40° + \mathrm{Q} + 135° = 180°\)

\(\mathrm{Q} = 180° - 175° = 5°\)

Since \(5°\) isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify the exterior angle but set up the wrong equation, thinking the exterior angle equals one interior angle instead of the sum of two remote interior angles.

They might write: \(135° = \angle\mathrm{Q}\), leading them to incorrectly select Choice D (95°) by coincidence, or get confused about which angle the exterior angle actually represents.

The Bottom Line:

Success requires clearly understanding what an exterior angle is and remembering that it equals the sum of the two non-adjacent interior angles, not just one angle. The exterior angle theorem provides the most direct path to the solution.