Supplementary, an adjective, refers to something added in order to complete something else, to make up a deficiency, to extend or strengthen an object, item, or idea: "A food allowance was supplementary to his weekly pay." "He took vitamins as a supplementary part of his diet." Complementary is closely related to supplementary in meaning but emphasizes not so much addition as it does completion or an essential addition: "In winning basketball, offense and defense are complementary." "In everyone's life, food and sleep are complementary." "As a scholar, Professor Auray insists that reading plays a complementary role in his life."
Dictionary of problem words and expressions. Harry Shaw. 1975.