Interior and Exterior Angles

Date: 11/01/2001 at 13:33:49
From: Anonymous
Subject: Geometry

I have calculated that the value of an angle in a quadrilateral is 
65 degrees. The other angles are 12, 104, and 27 degrees, and the sum 
is equal to 208 and not 360. Is this correct?

Date: 11/02/2001 at 12:02:33
From: Doctor Peterson
Subject: Re: Geometry

Hi,

I suspect that you have a non-convex quadrilateral, and one of your 
angles is not an INTERIOR angle. It's the interior angles that add up 
to 360 degrees.

In particular, if the shape looks something like

                   +
                / /
             / 65/
          /      /
       /12      /
    +------+    /
         104\27/
             \ /
              +

then the interior angle corresponding to 104 degrees is 360 - 104 = 
256, and the sum of interior angles is 12+256+27+65 = 360 as expected.

Now, how did I identify the 104 degree angle as the culprit?

Suppose in general that you have a quadrilateral with normal angles A, 
B, and C, but that angle D was measured on the outside, so that the 
interior angle is really 360-D. Then the sum of interior angles is

    A + B + C + (360-D) = 360

and

    A + B + C = D

To find which angle was measured on the outside, you can compare the 
sum you got with what you expected:

    sum = A+B+C+D = (A+B+C) + D = 2D

so the mismeasured angle is just half of the sum you got. Half of 208 
is 104, which is why I chose that as the reflex angle in my example.

It's fun being a Math Detective! Now make sure you always measure 
interior angles in the future, so I won't have to arrest you!

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