Angles Greater than 360 Degrees

Date: 01/01/99 at 09:41:21
From: Queenie
Subject: Angles -- Reflex, obtuse or acute?

Hi Dr. Math,

I know that a reflex angle measures between 180-360 degrees, an obtuse 
angle measures from 90-180 degrees, and an acute angle measures less 
than 90 degrees.

My question is, what is an angle that measures 425 degrees?

My class had a debate. Some people said that it should be a reflex 
angle because it's greater than 180 degrees, but some people said that 
it should be an acute angle because it went all the way around and 
over lapped, leaving an acute angle. I think I'm in for the acute 
angle, because when you draw an angle, to indicate that the angle is 
425 degrees instead of 65, you would have to draw the little arch all 
the way around and then end it. It looks quite weird. I'm just asking 
you to make sure.

Thanks for your time, looking forward to your reply! :-)
-Queenie

Date: 01/03/99 at 18:45:45
From: Doctor Pat
Subject: Re: Angles -- Reflex, obtuse or acute?

Queenie,

In mathematics we usually separate angles into "angles of inclination" 
and "angles of rotation." If you use the basic ideas of geometry in a 
plane, an angle is the "opening" between two rays. This leads to the 
names above. But if we talk about angles greater than 360 degrees, this 
can not happen "between" two rays. I have never heard anyone give 
either of the names to angles greater than 360 because we almost always 
are talking about the rotation of an angle in terms of some reference 
or stationary ray. Perhaps a more important term would be the term used 
in expressing the idea you gave when you wrote "because when you draw 
an angle, to indicate that the angle is 425 degrees instead of 65" is 
the word COTERMINAL. Mathematically we would say a 425 degree rotation 
is coterminal with a 65 degree rotation, and both are coterminal with 
a negative 295 degree rotation.  

Although I would not say a 425 degree angle is "acute," I would say it 
had an acute "reference angle." The purpose of the language is to help 
us understand the things which are alike, and those which are 
different, so to me, it wouldn't be accurate to just say a 425 degree 
angle is acute.  

Hope this helps.

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