Date: 05/29/2003 at 14:04:00 From: Lori Quinto-Green Subject: What to call two angles that add up to 360. We are teachers and we were discussing a problem that involved arc angles. We need the central angle that is not the angle of the sector. We need the rest of the angle. We were searching for a word to use to describe this angle. For example, an angle has a complement that totals 90, and a supplement that totals 180. What is the rest of the angle called when you want it to total 360?
Date: 05/30/2003 at 12:06:51 From: Doctor Peterson Subject: Re: What to call two angles that add up to 360. Hi, Lori. I tried searching for some of these terms to see if they are used, and found several discussions about the question: From the discussion group geometry-college, archived at the Math Forum: Angle1+Angle2=360 degrees -- RelationshipName? http://mathforum.org/kb/message.jspa?messageID=1071864 Ben Saucer: They're called explementary angles. Also called conjugate angles. An angle greater than 180 degrees, but less than 360 degrees is called a reflex angle. Pat Ballew: Here is what I know about the term "Explementary" copied directly from my web page at http://www.pballew.net/etyindex.html where there are now about 700 terms and their origins. Hope this helps.. ------------------------------------------------------------------ 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. The following discussion from 1996 and 1999 is the original exchange, on the discussion group geometry-pre-college, referred to above: What's the word? http://mathforum.org/kb/message.jspa?messageID=1075853 Here is the reference to "explementary" in a dictionary, which is mentioned in that discussion: Dictionary of Technical Terms for Aerospace Use - Daniel R. Glover http://roland.lerc.nasa.gov/~dglover/dictionary//content.html explement An angle equal to 360° 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°. Elsewhere in the same dictionary, http://roland.lerc.nasa.gov/~dglover/dictionary/a.html angle The inclination to each other of two intersecting lines, measured by the arc of a circle intercepted between the two lines forming the angle, the center of the circle being the point of intersection. An acute angle is less than 90°; a right angle 90 °; an obtuse angle, more than 90° but less than 180 °; a straight angle, 180°; a reflex angle, more than 180° but less than 360°; a perigon, 360°. Any angle not a multiple of 90° is an oblique angle. If the sum of two angles is 90°, they are complementary angles; if 180°, supplementary angles; if 360°, explementary angles. Two adjacent angles have a common vertex and lie on opposite sides of a common side. A dihedral angle is the angle between two intersecting planes. A spherical angle is the angle between two intersecting great circles. Another source is Maritime Safety Information Division (PDF document) http://pollux.nss.nima.mil/NAV_PUBS/APN/Chapt-21.pdf Two angles whose sum is a right angle (90°) are complementary angles, and either is the complement of the other. Two angles whose sum is a straight angle (180°) are supplementary angles, and either is the supplement of the other. Two angles whose sum is a circle (360°) are explementary angles, and either is the explement of the other. The two angles formed when any two lines terminate at a common point are explementary. So, though this term seems to be rare, it looks as if we've found our answer! I also looked up "conjugate angles", referred to by one poster, and found that it is present in several glossaries: Count On - Maths Year 2000 Dictionary http://www.mathsyear2000.org/dictionary/g_fset.html conjugate angles The conjugate of a given angle is the angle needed to make it up to 360 degrees (a whole turn). In the diagram, the red and green angles are each the conjugate angle of the other. For example, the conjugate angle to 100° is 260°. Angles and Measures http://www.geocities.com/mathfair2002/school/geo/geo0.htm Conjugate Angles Two angles are described as conjugate if they add up to 360° (2pi rad). e.g. the conjugate angle of 120° is given by 360° - 120° = 240° So we have not one, but two answers. I suspect that "conjugate" is used in too many other ways, and might lead to confusion in some contexts; so I'm inclined to go with "explementary," hard as it is to say. And, to repeat, the roots of the word mean "filling outside," which fits well. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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