To attract someone's attention and interest, or to promise to do something, or to join in an activity.
From Old French 'engager' meaning to pledge or promise, from 'en-' (in) plus 'gage' (pledge). Related to the noun 'gage' meaning a token.
When you get 'engaged' to be married, you're literally entering into a 'gage' or pledge—the same root appears in words like 'mortgage' (literally 'death pledge' in French), showing how financial and romantic promises use the same ancient language.
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