Most Students learn Supplementary and complimentary in geometry. They may even use the same mnemonic I do to remember which is which. 90 degree angles are crooked, (Complimentary ),180 degree angles are straight, (Supplementary).
Supplement The supplement of an angle is the angle that must be added to "fill up" a semi-circle. The sup root is a variation of the common sub for below or under. The ple is the same root that gives us the math word plus for "to increase or add to" something. Together they suggest the addition of something to fill the "low" amount. Several other English words are formed from the same roots. Supply is an alternative of the same word. The word supplicate, meaning beg or implore, if from one who needs to be supplied. Supple, for limber, is perhaps and early variation of "beggars can't be choosers"; those who need should remain flexible.
Jeff Miller's website gives the first use of the term (in this use) as:
In 1796 Hutton Math. Dict. has "The complement to 180° is usually called the supplement.
In 1798 Hutton in Course Math. has "supplemental arc" (one of two arcs which add to a semicircle) [OED].
He also gives"Supplement of a parallelogram" appears in English in 1570 in Sir Henry Billingsley’s translation of Euclid’s Elements. Maybe because the adjacent angles of a parallelogram are supplementary.
Complementary The Latin word complere means to complete. The ple root is the same root that gives us the word plus. Most mathematical uses of complement can be understood from this origin. A complement to an angle is the amount needed to complete a right angle. The "tens complement" of six is four, the amount that is needed to complete a ten. The word compliment, for an expression of praise or admiration, is from the same root. It came from a Spanish term for the gift that was given to repay someone for a favor. The gift that would complete the exchange.
Explementary I first heard of the word explementary in July of 1999. It was "re-created" by Steve Wells of a company called think3 while working on a new CAD program, thinkdesign. The word was needed to represent the angle required to complete a 360 degree circle. They wanted a word that would be a natural sounding extension of complement, and supplement. The Latin explementum means "filling" or "stuffing" (as reported by Ken Pledger, and other sources) and it is "explement" that is reported to be in the O.E.D. as "that which fills up". This is very much the same meaning as complement and supplement. After a couple of days, he found the word was not as new to mathematics as we had thought. Several days later he wrote to tell me that the word already appeared on the DICTIONARY OF TECHNICAL TERMS FOR AEROSPACE USE (Web edition edited by Daniel R. Glover, Jr.) NASA Lewis Research Center, Cleveland, Ohio. Here is their definition, as sent to me by Mr Wells:
"Explement -- An angle equal to 360 degrees minus a given angle. Thus, 150 ° is the explement of 210° and the two are said to be explementary. See complement, supplement.
Explementary angles -- Two angles whose sum is 360°."
My thanks to Mr Wells for his advice and corrections as much of this content came directly from his letters. But, it seems to have existed well before either of these. EXPLEMENT is found in 1827 in Mathematical and astronomical tables by William Galbraith: "...the explement, or difference from four right angles" [Google print search].
The term conjugate angles is also sometimes used. This may come from the polar representation of complex conjugates. Two complex numbers a + bi and a - bi are called conjugates, and the polar representations using the Argand diagram will have angles that sum to 360 degrees
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