In triangles LMN and RST, angles L and R each have measure 60°, LN = 10, and RT = 10....

1. TRANSLATE the problem information

• Given information:
  • \(\angle\mathrm{L} = \angle\mathrm{R} = 60°\) (one angle pair equal)
  • \(\mathrm{LN} = 10, \mathrm{RT} = 10\) (one side pair equal)
• Need: Additional information to prove triangle congruence

2. INFER the approach

• Since we have some angle information, let's check if we can establish all corresponding angles are equal (AAA similarity)
• Combined with our equal corresponding sides, this would prove congruence
• Strategy: Calculate the third angle in each triangle for each answer choice

3. SIMPLIFY angle calculations for each option

Option A & B: These give different side lengths for corresponding parts, so triangles can't be congruent.

Option C: \(\angle\mathrm{M} = 70°, \angle\mathrm{S} = 60°\)

• Triangle LMN: \(\angle\mathrm{N} = 180° - 60° - 70° = 50°\)
• Triangle RST: \(\angle\mathrm{T} = 180° - 60° - 60° = 60°\)
• Angles don't match up properly between triangles

Option D: \(\angle\mathrm{M} = 70°, \angle\mathrm{T} = 50°\)

• Triangle LMN: \(\angle\mathrm{N} = 180° - 60° - 70° = 50°\)
• Triangle RST: \(\angle\mathrm{S} = 180° - 60° - 50° = 70°\)

4. INFER the correspondence and congruence

For Option D:

• Triangle LMN has angles: 60°, 70°, 50°
• Triangle RST has angles: 60°, 70°, 50°
• Proper correspondence: \(\angle\mathrm{L} \leftrightarrow \angle\mathrm{R}, \angle\mathrm{M} \leftrightarrow \angle\mathrm{S}, \angle\mathrm{N} \leftrightarrow \angle\mathrm{T}\)
• All corresponding angles equal → AAA similarity
• \(\mathrm{LN} = \mathrm{RT} = 10\) (corresponding sides equal) → congruence proven

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't develop the strategy of calculating all three angles in both triangles to check for angle correspondence.

Instead, they might look at the given angle measures and immediately assume they know enough, or they might not realize they need to find the third angle in each triangle. Without calculating \(\angle\mathrm{N}\) and \(\angle\mathrm{S}\), they can't establish the proper correspondence between triangles. This leads to confusion about which answer choice actually provides sufficient information, causing them to guess or select an answer like Choice A or Choice B because the numbers "look different."

Second Most Common Error:

Conceptual confusion about congruence criteria: Students might not understand that AAA similarity combined with one pair of equal corresponding sides proves congruence.

They might think that having some equal angles and some equal sides isn't enough, or conversely, they might think that any combination of equal parts proves congruence without checking if the parts actually correspond properly. This may lead them to select Choice C because it mentions specific angle measures without verifying the complete angle correspondence.

The Bottom Line:

This problem requires systematic angle calculation and careful analysis of correspondence - students who jump to conclusions without complete analysis will struggle to identify the correct sufficient condition.