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]
(..) - This is a good one. - So... - Yeah? - Shuffle. Did you get her an engagement gift? We're not engaged, Dad. We're not together. We're just friends. All right, you don't wanna label it. I understand. But get her some jewelry. The only thing your mom loves is jewelry. I don't care how upset she is, (..)2011 Friends with Benefits