complement, compliment

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.

complement
  1. :: noun

    Something that completes, makes up a whole, or brings to perfection.

  2. :: noun

    The quantity or number needed to make up a whole: shelves with a full complement of books.

  3. :: noun

    Either of two parts that complete the whole or mutually complete each other.

  4. :: noun

    An angle related to another so that the sum of their measures is 90°.

compliment
  1. :: noun

    An expression of praise, admiration, or congratulation.

  2. :: noun

    A formal act of civility, courtesy, or respect.

  3. :: noun

    Good wishes; regards: Extend my compliments to your parents. See Usage Note at complement.

  4. :: verb-transitive

    To pay a compliment to.

Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.

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About Homophones

Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.

If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").