engaged has 7 distinct synonyms spread across 4 of its 7 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
having ones attention or mind or energy engaged
“she keeps herself fully occupied with volunteer activities”
(used of toothed parts or gears) interlocked and interacting
“the gears are engaged”
3 of the 7 recorded senses carry no synonym, antonym or related term, so they are not shown above. Every sense, with its definition and examples, is on the full entry for engaged.
Synonyms, antonyms and the broader and narrower relations for engaged come from Princeton University’s WordNet. Full sources and licences.