(heading) To interact socially.
(heading) To interact antagonistically.
(heading) To interact contractually.
(heading) To interact mechanically.
To enter into (an activity), to participate (construed with in).
To entangle.
That engages the attention; engrossing, interesting; enthralling.
I found the first of the Harry Potter books a very engaging read.
Charming; attractive, especially of a manner or behaviour.
Beauty, of course, and a bright, engaging personality — or at least the ability to fake one — are prerequisites for entering the Miss World competition.