Triangles PQR and XYZ are similar, where angle P corresponds to angle X. Angles Q and Y are right angles,...

1. TRANSLATE the problem information

• Given information:
  • Triangles PQR and XYZ are similar
  • P corresponds to X
  • Q and Y are both right angles (90°)
  • Angle P = 23°
  • Need to find angle Z

2. INFER the key relationships

• Since the triangles are similar, corresponding angles must be equal
• If P corresponds to X, then the remaining angles must correspond in order: Q↔Y and R↔Z
• This means: \(\mathrm{angle\;Z = angle\;R}\)

3. INFER the solution strategy

• To find angle Z, we need to find angle R first
• We can find angle R using the angle sum property in triangle PQR

4. SIMPLIFY to find angle R

• In triangle PQR: \(\mathrm{angle\;P + angle\;Q + angle\;R = 180°}\)
• Substituting known values: \(\mathrm{23° + 90° + angle\;R = 180°}\)
• Solving: \(\mathrm{angle\;R = 180° - 90° - 23° = 67°}\)

5. INFER the final answer

• Since angle Z corresponds to angle R: \(\mathrm{angle\;Z = 67°}\)

Answer: B (67 degrees)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret which angles correspond to each other, thinking that angle Z might correspond to angle P since P is given.

This leads them to conclude \(\mathrm{angle\;Z = 23°}\) and select Choice A (23 degrees).

Second Most Common Error Path:

Missing conceptual knowledge about similar triangles: Students remember that angles are related in similar triangles but forget that corresponding angles are equal, instead thinking that similar triangles must have the same angle measures in the same positions.

Without the correspondence concept, they get confused about which angle equals which, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students understand both the definition of similar triangles (corresponding angles are equal) and can correctly identify which angles correspond to each other based on the given information.