In triangle triangleABC, the measure of angle angleA is 50°. If triangle triangleABC is isosceles, which of the following is...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC is isosceles
  • Angle A measures \(50°\)
  • Need to find which angle B measure is NOT possible

• What this tells us: We need exactly two equal angles in our triangle

2. CONSIDER ALL CASES for isosceles triangles

Since an isosceles triangle has exactly two equal angles, and we know angle \(\mathrm{A} = 50°\), there are only three ways this can happen:

• Case 1: Angle A equals angle B (both \(50°\))
• Case 2: Angle A equals angle C (both \(50°\))
• Case 3: Angle B equals angle C (angle A is the different one)

3. INFER the angle measures for each case

Let's work out what all three angles would be in each case:

Case 1: \(\mathrm{A} = \mathrm{B} = 50°\)

• Angle C:
  \(\mathrm{C} = 180° - 50° - 50°\)
  \(\mathrm{C} = 80°\)
• Triangle angles: \(50°, 50°, 80°\) ✓

Case 2: \(\mathrm{A} = \mathrm{C} = 50°\)

• Angle B:
  \(\mathrm{B} = 180° - 50° - 50°\)
  \(\mathrm{B} = 80°\)
• Triangle angles: \(50°, 80°, 50°\) ✓

Case 3: \(\mathrm{B} = \mathrm{C} = \mathrm{x}\) (unknown)

• We have:
  \(50° + \mathrm{x} + \mathrm{x} = 180°\)
• Solving:
  \(50° + 2\mathrm{x} = 180°\)
  \(2\mathrm{x} = 130°\)
  \(\mathrm{x} = 65°\)
• Triangle angles: \(50°, 65°, 65°\) ✓

4. APPLY CONSTRAINTS to check each answer choice

Now I'll test each possible angle B value:

• Choice A (\(50°\)): This is Case 1 - valid ✓
• Choice B (\(65°\)): This is Case 3 - valid ✓
• Choice C (\(80°\)): This is Case 2 - valid ✓
• Choice D (\(100°\)): Let me check this...
  • If \(\mathrm{B} = 100°\), then
    \(\mathrm{C} = 180° - 50° - 100°\)
    \(\mathrm{C} = 30°\)
  • Triangle angles would be: \(50°, 100°, 30°\)
  • No two angles are equal → NOT isosceles ✗

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES reasoning: Students often only think about one configuration of the isosceles triangle, typically assuming the two equal angles must be the base angles. They might think "if \(\mathrm{A} = 50°\), then the other two angles must be equal" and immediately jump to \(\mathrm{B} = \mathrm{C} = 65°\).

With this limited thinking, they incorrectly believe that B can only equal \(65°\), leading them to select one of the other choices (A, B, or C) as the impossible option. This leads to confusion and guessing among the wrong answers.

Second Most Common Error:

Missing conceptual knowledge about isosceles triangles: Some students might confuse "isosceles" with "equilateral" and think all angles must be equal, or they might forget that isosceles means exactly two equal angles (not "at least two").

This conceptual confusion prevents them from properly setting up the cases, causing them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires systematic case analysis. Success depends on recognizing that there are multiple ways to arrange two equal angles in a triangle, then methodically checking each possibility rather than assuming only one configuration exists.