Greek-letter code to English words: If αδγηε corresponds to ARGUE and σφλπε to SOLVE, then παγηελω corresponds to which English word?

Introduction / Context:
We are shown a cipher where selected Greek letters consistently stand for specific English letters. Two examples are provided as keys; we must decode a third Greek sequence into an English word by consolidating the mapping from the examples and inferring any remaining letter by pattern (common vocabulary/orthography).

Given Data / Assumptions:

• αδγηε → ARGUE
• σφλπε → SOLVE
• Target: παγηελω

Concept / Approach:
Build a table from the two exemplars by aligning each Greek letter to its English counterpart at the same position. Reuse those mappings in the target. If a new Greek letter appears, infer it from the likely English word that results (guided by common English spelling and options provided).

Step-by-Step Solution:

Verification / Alternative check:
Check mutual consistency: all reused letters align with the established map; only ω is new, and mapping it to Y yields a dictionary word matching one of the choices without any contradiction.

Why Other Options Are Wrong:

• VAGRANT / VAGUER: Do not match the letter pattern V A G U E L ?
• VAGUELE: Not a standard English word; also duplicates E where the target shows ω (unknown, not E).
• None of these: Invalid because “VAGUELY” fits.

Common Pitfalls:
Assuming ω must match a vowel; overlooking that “VAGUELY” is a common adverb that completes the sequence naturally.

Final Answer:
VAGUELY