Definitions:

1. (of facilities such as telephones or lavatories) unavailable for use by anyone else or indicating unavailability adjective
   • (`engaged' is a British term for a busy telephone line)
2. (used of toothed parts or gears) interlocked and interacting adjective
   • the gears are engaged
3. built against or attached to a wall adjective
   • engaged columns
4. having services contracted for adjective
   • the carpenter engaged (or employed) for the job is sick
5. involved in military hostilities adjective
   • the desperately engaged ships continued the fight

Synonyms:

• booked
• busy
• intermeshed
• occupied
• set-aside