check, cheque

The words check, cheque sound the same but have different meanings and spellings. Why do check, cheque sound the same even though they are completely different words?

The answer is simple: check, cheque are homophones of the English language.

check
  1. :: noun

    An action or influence that stops motion or expression; a restraint: Heavy rains were a check on the army's advance.

  2. :: noun

    The condition of being stopped or held back; restraint: kept my temper in check; holding agricultural pests in check with sprays.

  3. :: noun

    An abrupt stop in forward movement or progress; a halt.

  4. :: noun

    The act or an instance of inspecting or testing, as for accuracy or quality; examination: the careful check of each unit before sale; gave the car an oil check.

cheque
  1. :: noun

    Chiefly British Variant of check.

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").