Quadrilateral P'Q'R'S' is similar to quadrilateral PQRS, where P, Q, R, and S correspond to P', Q', R', and S',...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P'Q'R'S' \sim PQRS}\) (similar quadrilaterals)
  • \(\mathrm{P \leftrightarrow P', Q \leftrightarrow Q', R \leftrightarrow R', S \leftrightarrow S'}\) (correspondence)
  • \(\mathrm{\angle P = 30°}\), \(\mathrm{\angle Q = 50°}\), \(\mathrm{\angle R = 70°}\)
  • Side scaling factor = 3

• What we need: measure of \(\mathrm{\angle P'}\)

2. INFER what similarity means for angles

• Key insight: Similar figures have identical corresponding angles
• The scaling factor affects side lengths, NOT angle measures
• Since P corresponds to P', we have \(\mathrm{\angle P = \angle P'}\)

3. Apply the angle correspondence

• \(\mathrm{\angle P = 30°}\) (given)
• \(\mathrm{\angle P = \angle P'}\) (corresponding angles in similar figures)
• Therefore: \(\mathrm{\angle P' = 30°}\)

Answer: B. 30°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students think the scaling factor affects angle measures

Many students see "each side is 3 times longer" and incorrectly assume this changes the angles too. They might think: "If the sides are 3 times bigger, maybe the angles are different too" or try to multiply 30° by 3 to get 90°.

This may lead them to select Choice D (90°) or cause confusion leading to guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students get overwhelmed by extra information and lose focus

The problem gives angles Q and R, plus the scaling factor - all irrelevant to finding \(\mathrm{\angle P'}\). Students might get distracted trying to use this extra information instead of focusing on the simple correspondence between P and P'.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

The key insight is recognizing that similarity preserves angle measures while only changing side lengths. The scaling factor is a red herring - once you know corresponding angles are equal, the answer is immediate.