Two symbol-to-letter decodings are given: 1) “@ # $ % &” decodes to “ARGUE”. 2) “α β γ δ &” decodes to “SOLVE”. Using these as keys, decode the symbol string: “δ @ $% &γ ∠”. What English word does it form?

Introduction / Context:
This item provides two known decodings (symbol sequences mapped to English words). From those, we infer individual symbol→letter mappings, then apply them to decode a new symbol string. The skill tested is inferring a cipher key from exemplars and applying it consistently.

Given Data / Assumptions:

• “@ # $ % &” → “A R G U E”.
• “α β γ δ &” → “S O L V E”.
• Query: decode “δ @ $% &γ ∠”.
• Assume single, consistent mapping; adjacency like “$%” means two symbols in sequence.

Concept / Approach:
Determine the per-symbol mapping from the two exemplars by aligning positions, then substitute for each symbol in the query. If any symbol appears only in the query (e.g., “∠”), infer the resulting word by dictionary sense-checking against options.

Step-by-Step Solution:

Verification / Alternative check:
Cross-check consistency: the partial decoding “VAGUEL” matches a valid English prefix; only “VAGUELY” fits the exact sequence and is offered as an option. No contradictions with prior mappings.

Why Other Options Are Wrong:

• VAGRANT: Would require R at the third position, but $ maps to G.
• VAGUELE: Nonstandard word; also final “E” would require “&”, not “∠”.
• VAGUER: Too short and mismatched ending relative to “γ ∠”.
• None of these: A correct, dictionary word consistent with the mapping exists (VAGUELY).

Common Pitfalls:
Reading groupings incorrectly (e.g., treating “$%” as one token); mixing earlier, unrelated symbol tables; overlooking that “γ” and “δ” are Greek letters, not Latin; ignoring dictionary validation to resolve a single unknown mapping.

Final Answer:
VAGUELY