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]
(..) Forgive me. How are you and Leonard? Oh, great. Everything's good. I see you got engaged. Yes, we did. Let me tell you a story. Dad! Sorry. I'm sure you won't grow (..)2014 TvShow: The Big Bang Theory Title: The Clean Room Infiltration Season: 8 Episode 11