Collocations for engaged - adj.

be

deeply, heavily
those who are deeply engaged in party politics

fully, totally
largely, mainly, primarily
actively, directly
He is actively engaged in strategic discussions.

busily
constantly
currently
intellectually, politically, socially
otherwise (formal)
I'm afraid Mr Wilson cannot see you now as he is otherwise engaged.

in
She was engaged in conversation with a client.

on
He is actively engaged on several projects.

with
We need to become more engaged with our history as a nation.
Mrs Scott is engaged with a customer at the moment. (BrE)

be
become, get
The couple got engaged last month.

happily
newly, recently

to
She's engaged to an actor.

engaged to be married

Engaged is used with these nouns:
couple, tone