The question being asked of me is the following:
What is the sum of a polygon's exterior angles?
Assuming, again, that the polygon is simple and convex, the answer I see repeatedly given is 360°. I'm wondering if there are some other criteria for defining an exterior angle that I'm ignorant of, because if we say only that an exterior angle is the angle around a vertex that is not the interior angle, or put another way, that an exterior angle is 360°-$\measuredangle interior$, centered at a vertex, how does the sum of all vertices' exterior angles = 360°?
An example
Let there be a drawing! ...or two:
Exterior angle: exterior angle
Am I defining exterior angle incorrectly? The most common illustrations employed essentially lob off 180° from every exterior angle so that they fit nicely together into a kind of pie chart, but no reasoning is given for ignoring 180° from each vertex. (I can't link to an example because I'm limited to two links.) The illustrations suppose that the polygon is shrunk to a size where the polygon becomes only a one-dimensional point, i.e. where the polygon is no longer a polygon and doesn't have even interior angles. Is there justification for this? My essential question is, if an interior angle = 30°, why isn't the exterior angle = 330°?
I'll use the following polygon as an example:
$\sum\measuredangle interior=a+b+c$
$\text{ }=(360-A)+(360-B)+(360-C)$
$\text{ }=3*360-(A+B+C)$
$\text{ }=1080-\sum\measuredangle exterior$
$\text{ }=180^\circ$ (a given, since it's a triangle)
$\Longrightarrow A+B+C=1080-180=900^\circ$
$\sum\measuredangle exterior=900^\circ$
General solution
For any n-sided polygon, with interior angles a, b, ... n:
$\sum\measuredangle interior=a+b+...+n$
$\text{ }=(360-A)+(360-B)+...+(360-N)$
$\text{ }=360n-(A+B+...+N)$
$\text{ }=360n-\sum\measuredangle exterior$
Using the equation for the sum of interior angles of an n-sided polygon, $(n-2)180$,
$(n-2)180=360n-\sum\measuredangle exterior$
$\mathbf{\Longrightarrow\sum\measuredangle exterior=(n+2)180}$
For the triangle, this also works out to 900°.
For a hexagon, for example:
$\sum\measuredangle exterior=6*180+360=1440^\circ$
Granted, these answers have nothing to do with whether my definition of exterior angle is correct, but they're simply the answers I would get.
Also, for n=0, such as when the polygon is shrunk down to a point, as in the illustrations previously mentioned, the equation gives the expected 360°.