In triangle DEF, the exterior angle at vertex D measures 130°. Triangle DEF is isosceles. Which of the following is...

1. TRANSLATE the problem information

• Given information:
  • Triangle DEF is isosceles (exactly two angles are equal)
  • Exterior angle at vertex D measures 130°
  • Need to identify which angle measure is NOT possible for angle F

• What this tells us: We need to find the interior angle at D first, then explore all possible configurations

2. INFER the approach

• Since we know the exterior angle, we can find the interior angle at D using supplementary angles
• For an isosceles triangle, we need to CONSIDER ALL CASES for which two angles could be equal
• We'll test each possibility systematically

3. Find the interior angle at D

• Interior angle at D = \(180° - 130° = 50°\)

4. CONSIDER ALL CASES for the isosceles property

Case 1: D = E = 50°

• If the two equal angles are D and E
• Then \(\mathrm{F} = 180° - 50° - 50° = 80°\)

Case 2: D = F = 50°

• If the two equal angles are D and F
• Then \(\mathrm{E} = 180° - 50° - 50° = 80°\)

Case 3: E = F (the two angles opposite to D are equal)

• INFER that we can use the exterior angle theorem here
• Exterior angle at D = E + F = 130°
• Since E = F, we have \(2\mathrm{F} = 130°\)
• Therefore \(\mathrm{F} = 65°\)

5. APPLY CONSTRAINTS to test F = 75°

• Check if F = 75° can work with isosceles property:
  • If D = F: \(50° \neq 75°\) ✗
  • If E = F = 75°: Then \(\mathrm{D} = 180° - 75° - 75° = 30°\), but \(\mathrm{D} = 50°\) ✗
  • This creates contradictions with our known information

6. INFER the final answer

• Possible values for angle F: 50°, 65°, 80°
• The value 75° leads to contradictions

Answer: C (75°)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often only consider one or two configurations of the isosceles triangle instead of systematically examining all possibilities. They might find that F = 80° works and assume that's the only possibility, leading them to incorrectly eliminate one of the valid options (50° or 65°). This incomplete analysis causes them to guess randomly among the remaining choices.

Second Most Common Error:

Poor INFER reasoning about exterior angle theorem: Students may struggle to connect the exterior angle theorem to the case where the two remote interior angles (E and F) are equal. They might attempt to solve this case using only the angle sum property, missing the direct relationship that exterior angle = E + F = 130°. This leads to confusion and potentially selecting Choice A (50°) or Choice D (80°) as impossible when they actually are possible.

The Bottom Line:

This problem challenges students to systematically explore multiple scenarios while managing several geometric relationships simultaneously. Success requires both methodical case analysis and strategic application of the exterior angle theorem.