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]
(..) - I got his back. Yeah, right. You're just jealous because it turns out I'm Penny's number two choice after Leonard. Hey, if I wasn't engaged to Bernadette, that totally could have been me. Please. Sheldon would have been before you, and he might not even (..)2011 TvShow: The Big Bang Theory Title: The Skank Reflex Analysis Season: 5 Episode 1
(..) Just let it go, Sheldon. If I could, I would, but I can't, so I shan't. Now, knowing Penny, the obvious answer is, they engaged in coitus. But... since that's what it looked like, we can rule that out. Let's put on (..)2011 TvShow: The Big Bang Theory Title: The Skank Reflex Analysis Season: 5 Episode 1