Explementary Angles

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/