Quadrilaterals ABCD and PQRS are congruent, where A corresponds to P, B corresponds to Q, C corresponds to R, and...

1. TRANSLATE the problem information

• Given information:
  • Quadrilaterals ABCD and PQRS are congruent
  • A↔P, B↔Q, C↔R, D↔S (correspondence)
  • In ABCD: \(\mathrm{\angle A} = 85°\), \(\mathrm{\angle B} = 110°\), \(\mathrm{\angle C} = 75°\)
  • Need to find: \(\mathrm{\angle S}\)
• What this tells us: Since the quadrilaterals are congruent, corresponding angles are equal, so \(\mathrm{\angle S} = \mathrm{\angle D}\)

2. INFER the solution approach

• We need \(\mathrm{\angle S}\), but we don't know \(\mathrm{\angle D}\) yet
• Strategy: Find \(\mathrm{\angle D}\) first using the angle sum property, then use correspondence
• Key insight: We have three of the four angles in quadrilateral ABCD

3. SIMPLIFY to find angle D

• Interior angles in any quadrilateral sum to 360°:

\(\mathrm{\angle A} + \mathrm{\angle B} + \mathrm{\angle C} + \mathrm{\angle D} = 360°\)

• Substitute known values:

\(85° + 110° + 75° + \mathrm{\angle D} = 360°\)

\(270° + \mathrm{\angle D} = 360°\)

\(\mathrm{\angle D} = 90°\)

4. INFER the final answer using correspondence

• Since D corresponds to S in congruent quadrilaterals:

\(\mathrm{\angle S} = \mathrm{\angle D} = 90°\)

Answer: 90

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't realize they need to find angle D first and try to work directly with the correspondence without having all necessary information. They might look at the given angles (85°, 110°, 75°) and try to relate them directly to angle S, leading to confusion about which angle corresponds to which. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need to find angle D using \(270° + \mathrm{\angle D} = 360°\), but make arithmetic errors. They might calculate \(360° - 270° = 270°\) instead of 90°, or make other computational mistakes. This may lead them to select an incorrect numerical answer.

The Bottom Line:

This problem requires students to think strategically about the order of operations—you can't use the correspondence relationship until you have the angle you're trying to match. The key insight is recognizing that finding angle D is the necessary first step.