Adjectives

(s)engaged, occupiedhaving ones attention or mind or energy engaged“she keeps herself fully occupied with volunteer activities”, “deeply engaged in conversation”
(s)engagedinvolved in military hostilities“the desperately engaged ships continued the fight”
(s)booked, engaged, set-asidereserved in advance
(s)busy, engaged, in use(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”
(s)engaged, meshed, intermeshed(used of toothed parts or gears) interlocked and interacting“the gears are engaged”, “meshed gears”, “intermeshed twin rotors”
(s)engagedhaving services contracted for“the carpenter engaged (or employed) for the job is sick”
(s)engagedbuilt against or attached to a wall“engaged columns”
(s)bespoken, betrothed, engagedpledged to be married“the engaged couple”