Merriam-Webster's Collegiate® Dictionary, 11th Edition
COMPLEMENT
I. noun see: full
Date: 14th century
1. something that fills up, completes, or makes perfect, the quantity, number, or assortment required to make a thing complete , one of two mutually completing parts ; counterpart ,
2. the angle or arc that when added to a given angle or arc equals a right angle in measure, the set of all elements that do not belong to a given set and are contained in a particular mathematical set containing the given set, a number that when added to another number of the same sign yields zero if the significant digit farthest to the left is discarded, the musical interval required with a given interval to complete the octave, an added word or expression by which a predication is made complete (as president in “they elected him president” and beautiful in “he thought her beautiful”), the thermolabile group of proteins in normal blood serum and plasma that in combination with antibodies causes the destruction especially of particulate antigens (as bacteria and foreign blood corpuscles),
II. verb
Date: 1602
intransitive verb to exchange formal courtesies, transitive verb to be complementary to , compliment