"she keeps herself fully occupied with volunteer activities"
"deeply engaged in conversation"
involved in military hostilities;
"the desperately engaged ships continued the fight"
reserved in advance
(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);
"her line is busy"
"receptionists' telephones are always engaged"
"the lavatory is in use"
"kept getting a busy signal"
(used of toothed parts or gears) interlocked and interacting;
"the gears are engaged"
"meshed gears"
"intermeshed twin rotors"
having services contracted for;
"the carpenter engaged (or employed) for the job is sick"