In pentagon ABCDE, the measure of angleA is 108°. If each of the other three angles (angleC, angleD, and angleE)...

1. TRANSLATE the problem information

• Given information:
  • Pentagon ABCDE with \(\angle\mathrm{A} = 108°\)
  • Each of \(\angle\mathrm{C}\), \(\angle\mathrm{D}\), and \(\angle\mathrm{E}\) is at least 105° (meaning \(\angle\mathrm{C} \geq 105°\), \(\angle\mathrm{D} \geq 105°\), \(\angle\mathrm{E} \geq 105°\))
  • Need to find possible values for \(\angle\mathrm{B}\)

2. INFER the approach

• Since we have constraints on four angles and need the fifth, we should use the total angle sum
• The key insight: find the maximum possible value for \(\angle\mathrm{B}\) by minimizing the other constrained angles

3. TRANSLATE the pentagon angle sum

• For any pentagon: Sum of interior angles = \((5-2) \times 180° = 540°\)
• So: \(\angle\mathrm{A} + \angle\mathrm{B} + \angle\mathrm{C} + \angle\mathrm{D} + \angle\mathrm{E} = 540°\)

4. SIMPLIFY to isolate the unknown angles

• Substitute \(\angle\mathrm{A} = 108°\):
  \(108° + \angle\mathrm{B} + \angle\mathrm{C} + \angle\mathrm{D} + \angle\mathrm{E} = 540°\)
• Subtract 108° from both sides:
  \(\angle\mathrm{B} + \angle\mathrm{C} + \angle\mathrm{D} + \angle\mathrm{E} = 432°\)

5. INFER the constraint strategy

• To find the maximum possible \(\angle\mathrm{B}\), we need the minimum possible sum of \(\angle\mathrm{C} + \angle\mathrm{D} + \angle\mathrm{E}\)
• Since each of these three angles is at least 105°, their minimum sum is \(3 \times 105° = 315°\)

6. APPLY CONSTRAINTS to find the bound

• From step 4: \(\angle\mathrm{B} + (\angle\mathrm{C} + \angle\mathrm{D} + \angle\mathrm{E}) = 432°\)
• Since \((\angle\mathrm{C} + \angle\mathrm{D} + \angle\mathrm{E}) \geq 315°\):
  \(\angle\mathrm{B} + 315° \leq 432°\)
  \(\angle\mathrm{B} \leq 117°\)

7. APPLY CONSTRAINTS to select the answer

• Check each choice against \(\angle\mathrm{B} \leq 117°\):
  • (A) \(116° \leq 117°\) ✓ Valid
  • (B) \(118° \gt 117°\) ✗ Invalid
  • (C) \(126° \gt 117°\) ✗ Invalid
  • (D) \(180° \gt 117°\) ✗ Invalid

Answer: A (116°)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "at least 105°" as "exactly 105°" and calculate \(\angle\mathrm{B} = 432° - 315° = 117°\), then become confused when 117° isn't among the answer choices. They may randomly guess or incorrectly think the problem has no solution.

Second Most Common Error:

Poor INFER reasoning: Students correctly set up the constraint but fail to recognize that they need to find the maximum possible value of \(\angle\mathrm{B}\) by using the minimum values of the other angles. Instead, they might try to work backwards from each answer choice without establishing the bound first. This leads to inefficient checking and potential confusion about which choices are actually possible.

The Bottom Line:

This problem requires careful interpretation of inequality language ("at least") and strategic thinking about how constraints on some variables affect the bounds of others. Success depends on translating the constraint correctly and recognizing the optimization strategy.