1.Definition: (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"
2.Definition: (used of toothed parts or gears) interlocked and interacting
"The gears are engaged"
3.Definition: built against or attached to a wall
4.Definition: having ones attention or mind or energy engaged
"Deeply engaged in conversation"
5.Definition: having services contracted for
"The carpenter engaged (or employed) for the job is sick"
6.Definition: involved in military hostilities
"The desperately engaged ships continued the fight"
7.Definition: reserved in advance