He won't be able to attend because he's 'otherwise engaged'. [=he is busy doing something else]
('US') He is 'engaged' in research. = ('Brit') He is 'engaged' on research.
She 'engaged' him 'in' conversation. [=she started a conversation with him]
(..) It's worth more than my car. - Just, just give me your number. I'll find it. Yeah I'll start looking for it right now. Cause I got nothing else to do. I was engaged once, but I'm single now. Bye Hope you make your flight. I know it's around here somewhere, she's on a plane, that's the beauty of it. (..)2009 Zombieland