In triangles LMN and RST, angles L and R each have measure 60°, LN = 10, and RT = 30....

1. TRANSLATE the problem information

• Given information:
  • Angles L and R each measure 60°
  • LN = 10, RT = 30
  • Need additional info to prove triangles LMN and RST are similar

2. INFER the approach

• Since we need to prove similarity, I should consider the three similarity conditions:
  • AAA (three pairs of congruent angles)
  • SAS (two proportional sides with included angle congruent)
  • SSS (three pairs of proportional sides)
• I'll test each answer choice to see which one works

3. INFER what each choice offers and test systematically

Testing Choice A: MN = 7 and ST = 7

• Check if sides are proportional: \(\mathrm{LN/RT = 10/30 = 1/3}\), but \(\mathrm{MN/ST = 7/7 = 1}\)
• Since \(\mathrm{1/3 ≠ 1}\), the sides aren't proportional
• This cannot prove similarity

Testing Choice B: MN = 7 and ST = 21

• Check proportions: \(\mathrm{LN/RT = 10/30 = 1/3}\) and \(\mathrm{MN/ST = 7/21 = 1/3}\) ✓
• For SAS similarity, need the included angles to be congruent
• Angle L is between sides LM and LN, angle R is between RS and RT
• But I have sides MN and LN (and ST and RT), so the given angles aren't included
• This cannot prove similarity using SAS

Testing Choice C: Angles M = 70°, S = 60°

• SIMPLIFY to find missing angles using angle sum property:
  • In triangle LMN: \(\mathrm{angle\ N = 180° - 60° - 70° = 50°}\)
  • In triangle RST: \(\mathrm{angle\ T = 180° - 60° - 60° = 60°}\)

4. INFER corresponding angles and check for AAA similarity

• Triangle LMN has angles: \(\mathrm{L = 60°, M = 70°, N = 50°}\)
• Triangle RST has angles: \(\mathrm{R = 60°, S = 60°, T = 60°}\)
• For triangles to be similar, corresponding angles must be equal
• But \(\mathrm{M = 70°}\) doesn't match any angle in RST that equals 70°
• This doesn't work for AAA similarity

Testing Choice D: Angles M = 70°, T = 50°

• SIMPLIFY to find missing angles:
  • In triangle LMN: \(\mathrm{angle\ N = 180° - 60° - 70° = 50°}\)
  • In triangle RST: \(\mathrm{angle\ S = 180° - 60° - 50° = 70°}\)

5. INFER the correspondence and verify AAA condition

• Triangle LMN: \(\mathrm{L = 60°, M = 70°, N = 50°}\)
• Triangle RST: \(\mathrm{R = 60°, S = 70°, T = 50°}\)
• Corresponding angles: \(\mathrm{L↔R\ (both\ 60°), M↔S\ (both\ 70°), N↔T\ (both\ 50°)}\)
• All three pairs of corresponding angles are congruent!
• This satisfies AAA similarity condition

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't systematically test similarity conditions or incorrectly identify corresponding angles.

Many students see that choices A and B involve side lengths and immediately try to use proportional sides without carefully checking if the given angle is included between those sides. They might select Choice B thinking the proportional sides (1/3 ratio) are sufficient, not realizing the angles must be included between those specific sides for SAS similarity.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating missing angles using the angle sum property.

For Choice C, students might miscalculate one of the missing angles (like getting angle T = 50° instead of 60°) and conclude the triangles are similar when they're not. This leads to selecting Choice C based on incorrect angle calculations.

The Bottom Line:

This problem requires methodical testing of similarity conditions and careful attention to which angles correspond between triangles. Students need to remember that for SAS similarity, the congruent angle must be included between the proportional sides, and for AAA similarity, all corresponding angles must be identified correctly.