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### 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

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/
```
Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry
High School Triangles and Other Polygons

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