Defining Exterior Angles of Polygons

Date: 06/24/2004 at 12:47:10
From: Jeri
Subject: exterior angles of triangle --What is the anterior angle??

I am having a discussion with my math teacher.  He thought the sum of
the exterior angles of a triangle was 900 (360*3 - 180).  I showed him
that the sum is actually 360.  However, he said he must have been
thinking of the anterior angles.  What is an anterior angle?  Please
help!  Thanks!

Date: 06/24/2004 at 13:21:37
From: Doctor Peterson
Subject: Re: exterior angles of triangle --What is the anterior angle??

Hi, Jeri.

I've never heard of an anterior angle, and don't find that phrase used 
anywhere on the web along with "polygon".  But I have often wondered 
if there is a name for the entire angle on the outside of a polygon, 
and have felt uncomfortable calling the supplement of the interior 
angle the exterior angle when it seems so much more natural to use 
that name for the "explement" (outside) of the interior angle.

Now I know why.

Searching the web for such names, I found that although most sites 
give the definition you and I know for "exterior angle", at least two 
call the whole outer angle "exterior":

  Polygon Properties
    http://www.math.com/tables/geometry/polygons.htm 

    This site defines 'exterior angle' as the angle formed by two
    adjacent sides outside the polygon.

  Exterior Angle
    http://mathworld.wolfram.com/ExteriorAngle.html 

    An exterior angle b of a polygon is the angle formed externally
    between two adjacent sides.  It is therefore equal to 2pi - a,
    where a is the corresponding internal angle between two adjacent
    sides.

Pictures on both pages make it clear what they mean.  The latter page 
goes on to say

    Confusingly, a bisector of an angle c is known as an exterior
    angle bisector, while a bisector of an angle b (which is simply
    a line oriented in the opposite direction as the interior angle
    bisector) is not given any special name.

This latter term agrees with the familiar definition of exterior 
angle, rather than with the definition he gives, and may either be the 
origin of the common usage at lower levels, or a result of confusion 
over it.

So we seem to have different definitions for the same word in 
different mathematical circles, but some leakage between the two. Your 
teacher's comment, and my uneasiness, are two instances of 
interference between the two worlds.

Bottom line: be sure you know what definition someone is using before 
you use the word!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/