Triangle ABC is similar to triangle XYZ such that A, B, and C correspond to X, Y, and Z, respectively....

1. TRANSLATE the problem information

• Given information:
  • \(\triangle \mathrm{ABC} \sim \triangle \mathrm{XYZ}\) (similar triangles)
  • Correspondence: \(\mathrm{A \leftrightarrow X, B \leftrightarrow Y, C \leftrightarrow Z}\)
  • \(\angle \mathrm{A} = 35°\), \(\angle \mathrm{B} = 70°\)
  • Need to find angle Z

• What this tells us: We know two angles in triangle ABC, and we need to find the corresponding angle to the unknown third angle.

2. INFER the solution strategy

• Key insight: Since we need angle Z, and Z corresponds to C, we must first find angle C in triangle ABC
• Then we can use the property that corresponding angles in similar triangles are equal

3. SIMPLIFY to find the missing angle in triangle ABC

• Apply the angle sum property: \(\angle \mathrm{A} + \angle \mathrm{B} + \angle \mathrm{C} = 180°\)
• Substitute known values: \(35° + 70° + \angle \mathrm{C} = 180°\)
• Solve: \(\angle \mathrm{C} = 180° - 35° - 70° = 75°\)

4. INFER the final answer using similarity

• Since \(\triangle \mathrm{ABC} \sim \triangle \mathrm{XYZ}\) with C corresponding to Z
• Corresponding angles in similar triangles are equal
• Therefore: \(\angle \mathrm{Z} = \angle \mathrm{C} = 75°\)

Answer: C (75°)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to directly relate the given angles (35° or 70°) to angle Z without realizing they need to find angle C first.

They might think 'since A corresponds to X, then angle Z must be 35°' or similar faulty reasoning. This may lead them to select Choice A (35°) by incorrectly assuming direct correspondence with a given angle.

Second Most Common Error:

Poor TRANSLATE reasoning: Students may misunderstand the correspondence notation and confuse which triangle angle corresponds to which.

They might calculate angle C correctly as 75° but then think it corresponds to a different angle in triangle XYZ, leading to confusion and potentially guessing among the wrong choices.

The Bottom Line:

This problem requires understanding that similarity problems often involve a two-step process: first complete the information in one triangle, then apply the similarity relationship. The key insight is recognizing that you can't directly jump to the answer without finding the missing piece first.