Polygons LMNOP and ABCDE are similar, where vertices L, M, and N correspond to A, B, and C, respectively. If...

1. TRANSLATE the problem information

• Given information:
  • Polygons LMNOP and ABCDE are similar
  • L corresponds to A, M to B, N to C
  • Exterior angle at vertex P = \(68°\)
  • \(\mathrm{LO} = 12\), \(\mathrm{AD} = 18\)
  • Need to find interior angle at vertex E

2. INFER the complete vertex correspondence

• Since we have 5-vertex polygons and know \(\mathrm{L}\rightarrow\mathrm{A}\), \(\mathrm{M}\rightarrow\mathrm{B}\), \(\mathrm{N}\rightarrow\mathrm{C}\), we can determine:
  • The remaining vertices must correspond in order: \(\mathrm{O}\rightarrow\mathrm{D}\) and \(\mathrm{P}\rightarrow\mathrm{E}\)
• This means vertex P in LMNOP corresponds to vertex E in ABCDE

3. INFER the angle relationship using similar polygon properties

• In similar polygons, corresponding angles are always congruent
• Since P corresponds to E: exterior angle at P = exterior angle at E
• Therefore: exterior angle at E = \(68°\)

4. INFER the interior angle using linear pair relationship

• Interior and exterior angles at any vertex form a linear pair (they're supplementary)
• Interior angle at E + exterior angle at E = \(180°\)
• Interior angle at E = \(180° - 68° = 112°\)

Answer: C) 112°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the complete vertex correspondence pattern. Students may see \(\mathrm{L}\rightarrow\mathrm{A}\), \(\mathrm{M}\rightarrow\mathrm{B}\), \(\mathrm{N}\rightarrow\mathrm{C}\) but fail to determine that \(\mathrm{P}\rightarrow\mathrm{E}\), instead thinking P might correspond to some other vertex. This confusion about which angles actually correspond leads to using the wrong angle relationships and selecting incorrect answers like Choice A (45°) or Choice D (135°).

Second Most Common Error:

Conceptual confusion: Mixing up interior and exterior angles. Students might correctly find that corresponding angles are congruent and determine the angle at E is \(68°\), but then select Choice B (68°) without realizing they found the exterior angle when the problem asks for the interior angle.

The Bottom Line:

This problem tests whether students can work systematically with vertex correspondence in similar polygons and distinguish between interior and exterior angles. The key insight is recognizing that similar polygons preserve both the correspondence pattern and all angle relationships.