In triangles DEF and JKL, angles D and J each have measure 40°, the length of side DF is 15,...

1. TRANSLATE the problem information

Given information:

• \(\angle\mathrm{D} = \angle\mathrm{J} = 40°\) (one pair of congruent angles already established)
• \(\mathrm{DF} = 15\), \(\mathrm{JL} = 20\) (specific side lengths)
• Need to find which additional information proves similarity

What this tells us: We already have one piece of similarity evidence, so we need to identify what type of additional evidence each choice provides.

2. INFER the triangle correspondence

Strategic reasoning: Based on the given congruent angles (\(\angle\mathrm{D} = \angle\mathrm{J}\)) and the pairing of sides (DF with JL), the correspondence must be:

• \(\mathrm{D} \leftrightarrow \mathrm{J}\), \(\mathrm{E} \leftrightarrow \mathrm{K}\), \(\mathrm{F} \leftrightarrow \mathrm{L}\)

What to do: Test each answer choice to see if it provides sufficient evidence for AA, SAS, or SSS similarity.

3. INFER which similarity criterion each choice attempts

Choice A (DE = 12, JK = 15): This gives sides adjacent to the known congruent angles, suggesting SAS similarity.

Choice B (EF = 12, KL = 16): This gives two pairs of proportional sides but not adjacent to the known angle, creating SSA.

Choices C & D: These provide additional angle measures, suggesting AA similarity.

4. SIMPLIFY by testing each choice systematically

Choice A - Testing SAS:

• For SAS, sides forming the congruent angles must be proportional
• \(\angle\mathrm{D}\) formed by sides DE and DF; \(\angle\mathrm{J}\) formed by sides JK and JL
• Check: \(\frac{\mathrm{DE}}{\mathrm{JK}} = \frac{12}{15} = \frac{4}{5}\), but \(\frac{\mathrm{DF}}{\mathrm{JL}} = \frac{15}{20} = \frac{3}{4}\)
• Since \(\frac{4}{5} \neq \frac{3}{4}\), SAS fails ✗

Choice B - Testing proportionality:

• \(\frac{\mathrm{KL}}{\mathrm{EF}} = \frac{16}{12} = \frac{4}{3}\), \(\frac{\mathrm{JL}}{\mathrm{DF}} = \frac{20}{15} = \frac{4}{3}\) ✓ (same ratio)
• But this creates SSA (Side-Side-Angle), which is NOT a valid similarity criterion ✗

Choice C - Testing AA:

• TRANSLATE to find all angles: If \(\angle\mathrm{E} = 100°\) and \(\angle\mathrm{K} = 30°\)
• Triangle DEF: \(\angle\mathrm{F} = 180° - 40° - 100° = 40°\)
• Triangle JKL: \(\angle\mathrm{L} = 180° - 40° - 30° = 110°\)
• Corresponding angles: \(\angle\mathrm{E} \neq \angle\mathrm{K}\) (\(100° \neq 30°\)) ✗

Choice D - Testing AA:

• TRANSLATE to find all angles: If \(\angle\mathrm{E} = 110°\) and \(\angle\mathrm{L} = 30°\)
• Triangle DEF: \(\angle\mathrm{F} = 180° - 40° - 110° = 30°\)
• Triangle JKL: \(\angle\mathrm{K} = 180° - 40° - 30° = 110°\)
• INFER the correspondence check:
  • \(\angle\mathrm{D} = \angle\mathrm{J} = 40°\) ✓
  • \(\angle\mathrm{E} = \angle\mathrm{K} = 110°\) ✓
  • \(\angle\mathrm{F} = \angle\mathrm{L} = 30°\) ✓

All three pairs of corresponding angles are congruent → AA similarity proven!

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning about similarity criteria: Students often confuse SSA (Side-Side-Angle) with valid similarity criteria.

In Choice B, they see that two pairs of sides are proportional (\(\frac{\mathrm{KL}}{\mathrm{EF}} = \frac{\mathrm{JL}}{\mathrm{DF}} = \frac{4}{3}\)) and incorrectly conclude this proves similarity. They fail to recognize that the known congruent angle is not between the proportional sides, making this SSA rather than SAS.

This may lead them to select Choice B (EF = 12 and KL = 16).

Second Most Common Error:

Poor TRANSLATE skills with angle relationships: Students may incorrectly interpret which angles correspond to which in the similarity comparison.

For example, in Choice D, they might compare \(\angle\mathrm{E}\) with \(\angle\mathrm{L}\) instead of recognizing that \(\angle\mathrm{E}\) corresponds to \(\angle\mathrm{K}\) and \(\angle\mathrm{L}\) corresponds to \(\angle\mathrm{F}\). This leads to incorrect angle comparisons and rejecting the right answer.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

Success requires both understanding the three valid similarity criteria (AA, SAS, SSS) AND correctly establishing which angles and sides correspond between the triangles. The most challenging aspect is recognizing that SSA is not a valid criterion, even when the ratios work out perfectly.