engage

engage /ɪnˈgeɪʤ/ verb
engages; engaged; engaging
engage
/ɪnˈgeɪʤ/
verb
engages; engaged; engaging
Britannica Dictionary definition of ENGAGE
1 
[+ object] formal : to hire (someone) to perform a particular service : to pay for (help, services, etc.)
2 
[+ object] : to get and keep (someone's attention, interest, etc.)
3 
: to start fighting against (an opponent)
[+ object]
[no object]
(formal) — opposite disengage
4 
: to move (a mechanism or part of a machine) so that it fits into another part
[+ object]
[no object]
— opposite disengage

engage in

[phrasal verb]
1 
engage in (something) : to do (something)
2 
engage (someone) in (something) : to cause (someone) to take part in (something)

engage with

[phrasal verb]
engage with (someone or something) formal
: to give serious attention to (someone or something)
: to become involved with (someone or something)