When used as a verb, compliment means to offer commendation or praise.
The coach complimented the team for its efforts.
As an adjective, compliment refers to something provided at no cost.
Each case of wine includes a complimentary set of glasses.
Complement means to augment a process or complete a group.
The beige carpet complements the new wallpaper nicely.
The ship had its full complement of 150 sailors.
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