Dictionary.net

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"

involved in military hostilities; "the desperately engaged ships continued the fight"

built against or attached to a wall; "engaged columns"

reserved in advance

(architecture) built against or attached to a wall; "engaged columns"

(used of toothed parts or gears) interlocked and interacting; "the gears are engaged"; "meshed gears"; "intermeshed twin rotors"

involved in military hostilities; "the desperately engaged ships continued the fight"

of Engage

Occupied; employed; busy.

Pledged; promised; especially, having the affections pledged; promised in marriage; affianced; betrothed.

Greatly interested; of awakened zeal; earnest.

of Engage

Occupied; employed; busy.

Pledged; promised; especially, having the affections pledged; promised in marriage; affianced; betrothed.

Greatly interested; of awakened zeal; earnest.

Busy or occupied; affianced or betrothed.

Affianced.

Occupied or busy.