The sum of the measures of the interior angles of a polygon with n sides is given by the formula...

1. TRANSLATE the problem information

• Given information:
  • Pentagon (5-sided polygon)
  • Four interior angles: 98°, 107°, 116°, 123°
  • Need to find the fifth interior angle

2. INFER the solution strategy

• Key insight: The sum of ALL interior angles is fixed for any pentagon
• Strategy: Find total sum, then subtract the four known angles to get the fifth

3. TRANSLATE the polygon formula

• For any n-sided polygon: Sum of interior angles = \((n-2) \times 180°\)
• For pentagon: \(n = 5\), so sum = \((5-2) \times 180° = 3 \times 180° = 540°\)

4. SIMPLIFY by adding the known angles

• Sum of four given angles: \(98° + 107° + 116° + 123° = 444°\) (use calculator)

5. SIMPLIFY to find the missing angle

• Fifth angle = Total sum - Sum of known angles
• Fifth angle = \(540° - 444° = 96°\)

Answer: A (96)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students calculate the sum of the four given angles (444°) but then select this as their final answer instead of recognizing they need to subtract it from the total sum.

They think: "The problem asks for an angle measure, and 444° is the sum I calculated from the given angles."

This leads them to select Choice C (444).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or incorrectly apply the interior angle sum formula, leading to confusion about what the total should be.

They might calculate the total sum correctly as 540° but then think this IS the measure of the fifth angle.

This causes them to select Choice D (540).

The Bottom Line:

This problem tests whether students can connect the polygon angle sum formula with the logical step of finding a missing part when given the total and other parts. Success requires both knowing the formula AND recognizing the subtraction strategy.