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### 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

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/
```
Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry
Middle School Definitions
Middle School Two-Dimensional Geometry

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