Right triangle ABC with right angle at C is congruent to right triangle DEF with right angle at F, where...

1. TRANSLATE the problem information

• Given information:
  • Right triangle ABC ≅ right triangle DEF
  • Angle A corresponds to angle D
  • \(\mathrm{cos(A) = \frac{21}{29}}\)
  • Need to find \(\cos(\mathrm{D})\)

2. INFER the key relationship

• Since the triangles are congruent, corresponding angles must be equal in measure
• If angle A corresponds to angle D, then \(\mathrm{\angle A = \angle D}\)

3. INFER the trigonometric connection

• When two angles have the same measure, their trigonometric functions are identical
• Since \(\mathrm{\angle A = \angle D}\), then \(\mathrm{cos(A) = cos(D)}\)
• Therefore: \(\mathrm{cos(D) = \frac{21}{29}}\)

Answer: B) \(\mathrm{\frac{21}{29}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect triangle congruence to trigonometric function equality

Students may understand that triangles are congruent but fail to realize this means corresponding angles are equal. They might try to calculate \(\mathrm{cos(D)}\) using the Pythagorean theorem or attempt complex trigonometric calculations instead of recognizing the direct relationship. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Missing conceptual knowledge about congruent triangles: Students may not fully understand what "corresponding angles" means

Some students might think that congruent triangles only have equal sides, forgetting about equal angles. Or they might not understand which parts correspond to each other. This conceptual gap prevents them from making the connection that \(\mathrm{\angle A = \angle D}\), leading them to attempt unnecessary calculations and potentially select Choice A (\(\mathrm{\frac{20}{29}}\)) if they confuse cosine with sine relationships.

The Bottom Line:

This problem tests whether students can connect geometric concepts (triangle congruence) with trigonometric functions. The key insight is recognizing that equal angles have equal trigonometric values - no complex calculations needed!