Triangles ABC and DEF are similar, where A, B, and C correspond to D, E, and F, respectively. The measure...

1. TRANSLATE the problem information

• Given information:
  • Triangles ABC and DEF are similar
  • A, B, and C correspond to D, E, and F, respectively
  • \(\angle\mathrm{B} = 65°\)
  • \(\mathrm{AC} = 21\) and \(\mathrm{DF} = 7\)
  • Need to find: \(\angle\mathrm{E}\)

2. INFER the correct approach

• Since we need to find an angle measure in similar triangles, we should focus on the angle relationships
• The correspondence tells us: \(\mathrm{A}\leftrightarrow\mathrm{D}\), \(\mathrm{B}\leftrightarrow\mathrm{E}\), \(\mathrm{C}\leftrightarrow\mathrm{F}\)
• Similar triangles have equal corresponding angles

3. Apply the corresponding angles property

• Since B corresponds to E: \(\angle\mathrm{E} = \angle\mathrm{B}\)
• Therefore: \(\angle\mathrm{E} = 65°\)

Answer: D) \(65°\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students get distracted by the side length information (\(\mathrm{AC} = 21\), \(\mathrm{DF} = 7\)) and think they need to use it somehow. They might try to find a scale factor or set up proportions, completely forgetting that angle measures in similar triangles are simply equal to their corresponding angles.

This leads to confusion and abandoning the systematic approach, often resulting in guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the correspondence statement and think that since \(\mathrm{AC} = 21\) and \(\mathrm{DF} = 7\), somehow the angles are related by the same ratio (\(21:7 = 3:1\)). They might calculate \(65° \div 3 \approx 21°\) or try other incorrect numerical relationships.

This may lead them to select Choice B (\(21°\)).

The Bottom Line:

This problem tests whether students can distinguish between the two key properties of similar triangles: corresponding angles are equal (not proportional) while corresponding sides are proportional (not equal). The side length information is a red herring designed to see if students stay focused on what they actually need.