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]
(..) You tell me. Orson... I thought we... Orson. Oh, I just assumed now that we're engaged... Well, does that change our core values? No. But when I saw you standing in front of the freezer with a clingy silk dress, (..)2006 TvShow: Desperate Housewives Title: Listen to the Rain on the Roof Season: 3 Episode 1
(..) If you don't tell us, we're not coming. Oh, all right. I wanted to wait and make a proper announcement, but... Orson and I are engaged. Oh, my Gosh! Oh, let me see! You move fast. I know it's quick, but it just feels so right. (..)2006 TvShow: Desperate Housewives Title: Listen to the Rain on the Roof Season: 3 Episode 1