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The adjective "complementary" comes from the Latin complementum, which derives from the fourth form of the Latin verb (compleo:I fill up, complere:to fill up, complevi:I filled up, completum: having been filled up). An acute angle is filled up by its complement to form a right angle, which is the most basic kind of angle, because it can be constructed by bisecting a line. A right angle is a complete (full) angle.
See also: supplementary angles.