In convex pentagon PQRST, angle P is congruent to angle Q, and angle R is congruent to angle S. The...

1. TRANSLATE the problem information

• Given information:
  • Pentagon PQRST is convex
  • \(\angle\mathrm{P} \cong \angle\mathrm{Q}\), and \(\angle\mathrm{P} = 118°\)
  • \(\angle\mathrm{R} \cong \angle\mathrm{S}\) (unknown measures)
  • \(\angle\mathrm{T} = 96°\)

• What this tells us:
  • Since \(\angle\mathrm{P} \cong \angle\mathrm{Q}\), then \(\angle\mathrm{Q} = 118°\) also
  • Since \(\angle\mathrm{R} \cong \angle\mathrm{S}\), we can call both of these unknown angles \(\mathrm{x}\)

2. INFER the solution approach

• We have a 5-sided polygon with some known and unknown angle measures
• The key insight: All interior angles must sum to a specific total
• Strategy: Use the pentagon angle sum formula, then solve for the unknown angles

3. Apply the interior angle sum formula

• For any n-sided polygon: \(\mathrm{sum} = (\mathrm{n}-2) \times 180°\)
• For a pentagon \((\mathrm{n} = 5)\):
  \(\mathrm{sum} = (5-2) \times 180°\)
  \(= 3 \times 180°\)
  \(= 540°\)

4. TRANSLATE our angle information into an equation

• All five angles must sum to 540°:
  \(\angle\mathrm{P} + \angle\mathrm{Q} + \angle\mathrm{R} + \angle\mathrm{S} + \angle\mathrm{T} = 540°\)

• Substituting known values:
  \(118° + 118° + \mathrm{x} + \mathrm{x} + 96° = 540°\)

5. SIMPLIFY to solve for x

• Combine like terms: \(332° + 2\mathrm{x} = 540°\)
• Subtract 332° from both sides: \(2\mathrm{x} = 208°\)
• Divide by 2: \(\mathrm{x} = 104°\)

Answer: B (104)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "angle P is congruent to angle Q" into the mathematical understanding that both angles have the same measure. They may set up their equation incorrectly, perhaps writing \(\angle\mathrm{Q}\) as an unknown variable instead of recognizing it equals 118°.

This leads to an incorrect equation like: \(118° + \mathrm{y} + \mathrm{x} + \mathrm{x} + 96° = 540°\), where they now have two unknowns instead of one. This causes confusion and typically leads to guessing among the answer choices.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or incorrectly apply the interior angle sum formula for polygons. They might use 360° (thinking of exterior angles) or guess at other totals like 720°.

Using 360° would give:
\(118° + 118° + \mathrm{x} + \mathrm{x} + 96° = 360°\)
leading to \(332° + 2\mathrm{x} = 360°\)
so \(2\mathrm{x} = 28°\)
and \(\mathrm{x} = 14°\).
Since 14° isn't among the choices, this leads them to select Choice A (96) by guessing or assuming they made a calculation error.

The Bottom Line:

This problem tests whether students can systematically translate angle relationships into equations and apply the correct polygon formula. Success requires both precise language interpretation and geometric formula recall.