Etymology
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engaged (adj.)

"affianced, betrothed," 1610s, past-participle adjective from engage. Of telephone lines from 1891.

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Definitions of engaged

engaged (adj.)
reserved in advance;
Synonyms: booked / set-aside
engaged (adj.)
(of facilities such as telephones or lavatories) unavailable for use by anyone else or indicating unavailability; (`engaged' is a British term for a busy telephone line);
receptionists' telephones are always engaged
Synonyms: busy / in use
engaged (adj.)
having ones attention or mind or energy engaged;
deeply engaged in conversation
Synonyms: occupied
engaged (adj.)
involved in military hostilities;
the desperately engaged ships continued the fight
engaged (adj.)
(used of toothed parts or gears) interlocked and interacting;
the gears are engaged
Synonyms: meshed / intermeshed
engaged (adj.)
having services contracted for;
the carpenter engaged (or employed) for the job is sick
engaged (adj.)
built against or attached to a wall;
engaged columns
From wordnet.princeton.edu