Triangle PQR is isosceles with PQ = PR. If the exterior angle at vertex P measures 96°, what is the...

1. TRANSLATE the problem information

• Given information:
  • Triangle PQR is isosceles with \(\mathrm{PQ = PR}\)
  • Exterior angle at vertex P measures 96°
  • Need to find measure of angle Q

2. INFER the key relationship

• Since \(\mathrm{PQ = PR}\), triangle PQR is isosceles with P as the vertex angle
• This means angles Q and R are the base angles, so they're equal: \(\mathrm{∠Q = ∠R}\)

3. INFER the strategy using exterior angle property

• An exterior angle equals the sum of the two remote interior angles
• The exterior angle at P equals \(\mathrm{∠Q + ∠R}\)
• Since \(\mathrm{∠Q = ∠R}\), if we call each base angle x, then: \(\mathrm{96° = x + x = 2x}\)

4. SIMPLIFY to find the answer

• \(\mathrm{2x = 96°}\)
• \(\mathrm{x = 96° ÷ 2 = 48°}\)
• Therefore \(\mathrm{∠Q = 48°}\)

Answer: 48

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize which angles are equal in the isosceles triangle. They might think all three angles are equal (confusing isosceles with equilateral) or incorrectly identify which angles are the base angles.

This conceptual confusion leads them to set up incorrect equations, such as thinking \(\mathrm{∠P = ∠Q = ∠R}\), which would give them \(\mathrm{96° = 3x}\), leading to \(\mathrm{x = 32°}\). This causes them to get stuck or guess randomly.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember the exterior angle theorem and instead try to work only with the linear pair property and triangle angle sum. While this approach can work, it requires more steps and students often make calculation errors in the multi-step process.

Without the direct exterior angle relationship, they may incorrectly assume relationships between the interior angles, leading to confusion and incorrect calculations.

The Bottom Line:

This problem tests whether students can connect multiple geometric concepts: isosceles triangle properties, exterior angles, and angle relationships. The key insight is recognizing that the equal base angles can be treated as a single variable in the exterior angle equation.