Complement can be a noun or a verb and it basically means something that completes or makes something better, or to add to something in a way that improves it or makes it better.
“This wine is a nice complement to this cheese.”
“Those colors complement each other nicely.”
Compliment is a noun or a verb, which means a remark that expresses approval, admiration or respect or to say something nice that shows your approval, admiration or respect.
“When he told me the color of my shirt complemented my eyes, I thought it was such a nice compliment!”
The adjective forms are complementary and complimentary, but be careful! Complimentary has 2 meanings: free (a complimentary breakfast) and describing a situation in which someone gives you praise (he made some very complimentary comments).
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