When two people are engaged, they have agreed to marry each other. We got engaged on my eighteenth birthday
involved in military hostilities; "the desperately engaged ships continued the fight"
(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"
Someone who is engaged in or engaged on a particular activity is doing that thing. They found the three engaged in target practice. the various projects he was engaged on
Actively fighting one or more opponents in melee combat (Held, unconscious, or otherwise immobilized characters are not considered engaged unless they are actually being attacked )
involved in military hostilities; "the desperately engaged ships continued the fight
If a telephone or a telephone line is engaged, it is already being used by someone else so that you are unable to speak to the person you are phoning. The line is engaged
having ones attention or mind or energy engaged; "she keeps herself fully occupied with volunteer activities"; "deeply engaged in conversation"