The verb engage has 10 senses (first 6 from tagged texts)
1. (31) prosecute, engage, pursue -- (carry out or participate in an activity; be involved in; "She pursued many activities"; "They engaged in a discussion")
2. (9) absorb, engross, engage, occupy -- (engage or engross wholly; "Her interest in butterflies absorbs her completely")
3. (2) hire, engage, employ -- (engage or hire for work; "They hired two new secretaries in the department"; "How many people has she employed?")
4. (1) engage -- (ask to represent; of legal counsel; "I'm retaining a lawyer")
5. (1) betroth, engage, affiance, plight -- (give to in marriage)
6. (1) engage -- (get caught; "make sure the gear is engaged")
7. engage, wage -- (carry on (wars, battles, or campaigns); "Napoleon and Hitler waged war against all of Europe")
8. engage, enlist -- (hire for work or assistance; "engage aid, help, services, or support")
9. lease, rent, hire, charter, engage, take -- (engage for service under a term of contract; "We took an apartment on a quiet street"; "Let's rent a car"; "Shall we take a guide in Rome?")