complement, compliment - verb
Complement: to complete, to add to something, to provide something felt to be lacking, as in The two books complemented each other and provided a complete picture of the war.
Compliment: to praise or extol, as in She complimented him on his performance.
Complement: something that completes or makes perfect, as in The fine red wine was the perfect complement to the aged steak.
Compliment: an expression of praise or admiration, as in Her boss’s compliment raised her spirits.
Here’s a discussion from Vocabulary.com:
To complement, with an e, means to complete or supplement something, such as chocolates complementing the flowers you give your date. A complement is the thing doing the completing. The chocolates are a complement to the flowers. Here are some things that go well together:
“Even as teenagers, they finished each other's sentences, complementing and encouraging the other.” (Golf Digest) [Shouldn't this read complimenting?]
“In Ms. Clinton's case, IAC said her ‘skills and background complement the existing areas of expertise of other board members.’” (New York Times)
“GM mosquito technology must be evaluated as a complement to existing control measures.” (Scientific American)
To compliment, with an i, is to offer praise to or admiration. This could be in the form of words or actions. You might complimentyour date on his dance moves, for example. "Your moonwalk is perfect!" is a compliment. Here are more:
"I lived on people's compliments, kind words," she says. (Washington Post)
Biggest compliment: After Rob completed his waltz, Judge Len Goodman announced, "You've got the best footwork of any guy I've seen on this show ever." (Time)