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]
(..) will you comment... Secretary Hennings... Okay. Hello. Why haven't you told your mother you're engaged? Have you thought about children? I'm sorry we're late. That shade of lipstick does nothing for you. What kept you? (..)2002 Sweet Home Alabama
(..) some of the streets in Brooklyn. Maybe they'll disappear into a pothole. Your son's relationship with Melanie Carmichael... sources say they're engaged. Do you have a comment? Trust me, if my son were engaged, I'd have a comment. I thought you were tired. Jimmy, can you excuse us (..)2002 Sweet Home Alabama