Triangles ABC and DEF are congruent, where A corresponds to D, and B and E are right angles. The measure...

1. TRANSLATE the problem information

• Given information:
  • Triangles ABC and DEF are congruent
  • A corresponds to D
  • Angles B and E are right angles \(90°\)
  • Angle A = \(18°\)
  • Need to find angle F

• What this tells us: We can use the fact that corresponding angles in congruent triangles are equal.

2. INFER the correspondence pattern

• Since A corresponds to D, and B corresponds to E (both are right angles), then C must correspond to F
• Strategy: Find angle C first using the triangle angle sum, then use correspondence to find angle F

3. SIMPLIFY to find angle C

• In triangle ABC: \(\mathrm{Angle\ A + Angle\ B + Angle\ C} = 180°\)
• Substitute known values: \(18° + 90° + \mathrm{angle\ C} = 180°\)
• Combine: \(108° + \mathrm{angle\ C} = 180°\)
• Solve: \(\mathrm{angle\ C} = 180° - 108° = 72°\)

4. INFER the final answer

• Since angle C corresponds to angle F in congruent triangles
• \(\mathrm{angle\ F = angle\ C} = 72°\)

Answer: B. 72°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may misunderstand the correspondence and think that angle F directly corresponds to angle A since both are mentioned in the problem.

This leads them to conclude \(\mathrm{angle\ F} = 18°\), causing them to select Choice A (18°).

Second Most Common Error:

Inadequate INFER reasoning: Students may recognize they need to use the triangle angle sum but apply it to the wrong triangle or get confused about which angles they're looking for.

Some might think they need \(90°\) (the right angle measure) and select Choice C (90°), or add angles incorrectly and get Choice D (162°).

The Bottom Line:

The key challenge is recognizing that finding angle F requires an indirect approach—you must first find its corresponding angle C using the triangle angle sum property, then apply the congruence relationship.