In triangle PQR, the measure of the exterior angle at vertex P is 110°. If triangle PQR is isosceles, which...

1. TRANSLATE the problem information

• Given information:
  • Exterior angle at vertex P = \(110°\)
  • Triangle PQR is isosceles
• Need to find: Which measure is NOT possible for angle Q

2. INFER the starting point

• Since exterior and interior angles are supplementary, we need the interior angle at P first
• Interior angle at P = \(180° - 110° = 70°\)

3. CONSIDER ALL CASES for the isosceles triangle

• An isosceles triangle has exactly two equal angles
• There are three ways this can happen in triangle PQR:

Case 1: P is the vertex angle (Q = R)

• Let \(\angle Q = \angle R = x\)
• Sum: \(70° + x + x = 180°\)
• \(2x = 110°\), so \(x = 55°\)
• Therefore: \(\angle Q = 55°\)

Case 2: Q is the vertex angle (P = R)

• Since \(P = 70°\), then \(R = 70°\)
• \(\angle Q = 180° - 70° - 70° = 40°\)
• Therefore: \(\angle Q = 40°\)

Case 3: R is the vertex angle (P = Q)

• Since \(P = 70°\), then \(Q = 70°\)
• Therefore: \(\angle Q = 70°\)

4. APPLY CONSTRAINTS to identify the impossible measure

• Possible values for angle Q: \(40°\), \(55°\), \(70°\)
• Checking answer choices:
  • A. \(40°\) ✓ (from Case 2)
  • B. \(55°\) ✓ (from Case 1)
  • C. \(70°\) ✓ (from Case 3)
  • D. \(80°\) ✗ (impossible)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES reasoning: Many students assume that in an isosceles triangle, the given angle (P = \(70°\)) must be the vertex angle, so they only work out Case 1 where Q = R = \(55°\).

When they see that \(55°\) is answer choice B, they think it's the only possible value and incorrectly eliminate the other choices. This leads to confusion when trying to identify what's NOT possible, and they may randomly select Choice A (\(40°\)) or Choice C (\(70°\)) instead of recognizing these are actually valid.

Second Most Common Error:

Missing conceptual knowledge about isosceles triangles: Some students think "isosceles" means the triangle has one specific configuration, not realizing that ANY two angles can be equal. They get stuck trying to figure out which two angles should be equal and end up guessing.

The Bottom Line:

This problem tests whether students can systematically explore all possible configurations rather than jumping to the first solution they find. The key insight is recognizing that an isosceles triangle is defined by having exactly two equal angles, but those equal angles can be any pair of the three vertices.