1. To interact socially.
    1. To engross or hold the attention of; to keep busy or occupied.
      1. To draw into conversation.
      2. To attract, to please; to fascinate or win over (someone).
  1. To interact antagonistically.
  1. To enter into conflict with (an enemy).
    1. To enter into battle.
  1. To interact contractually.
    1. To arrange to employ or use (a worker, a space, etc).
      1. To guarantee or promise (to do something).
    2. To bind through legal or moral obligation (to do something, especially to marry) .
    3. To pledge, pawn (one's property); to put (something) at risk or on the line; to mortgage (houses, land).
      1. To interact mechanically.
    4. To mesh or interlock (of machinery, especially a clutch).
    5. To come into gear with.
      The teeth of one cogwheel engage those of another.
  2. To enter into (an activity), to participate (construed with ).

The above text is a snippet from Wiktionary: engage
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