The words complement, compliment sound the same but have different meanings and spellings. Why do complement, compliment sound the same even though they are completely different words?
The answer is simple: complement, compliment are homophones of the English language.
Something that completes, makes up a whole, or brings to perfection.
The quantity or number needed to make up a whole: shelves with a full complement of books.
Either of two parts that complete the whole or mutually complete each other.
An angle related to another so that the sum of their measures is 90°.
An expression of praise, admiration, or congratulation.
A formal act of civility, courtesy, or respect.
Good wishes; regards: Extend my compliments to your parents. See Usage Note at complement.
To pay a compliment to.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.