In a way that captures attention and interests someone; done in a manner that draws people in.
From 'engaging' (present participle of 'engage', from Old French 'engager') plus the adverbial suffix '-ly'. 'Engage' comes from Old French 'en-' (to cause to be) plus 'gage' (pledge), originally meaning to pledge or stake something.
Medieval knights would 'engage' by pledging their word and staking their honor in battle, so the word originally meant a serious commitment—now we use it to describe how a good story or speaker 'pledges' to keep your attention!
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