In the figure above check "regular". As you can see, for regular polygons all the exterior angles are the same, and like all polygons they add to 360° (see note below). So each exterior angle is 360 divided by the n, the number of sides. As a demonstration of this, drag any vertex towards the center of the polygon. You will see that the angles combine to a full 360° circle.
In the case of convex polygons, where all the vertices point "outwards" away form the interior, the exterior angles are always on the outside of the polygon. In the figure above, check "regular" and notice that this is the case.
Although there are two possible exterior angles at each vertex (see note below) we usually only consider one per vertex, selecting the ones that all go around in the same direction, clockwise in the figure above.
In the figure above click on 'reset' and check "both". You will see that both angles at each vertex are always congruent (same measure). This is because they form a pair of vertical angles, which are always congruent. Drag the vertices around and convince yourself this is true.
If the polygon is concave, things are a little trickier. A concave polygon has one or more vertices "pushed in" so they point towards the interior. In the figure above, uncheck "regular" and drag a vertex in towards the interior of the polygon. Notice that the exterior angle flips over into the inside of the polygon and becomes negative. If you add the exterior angles like before, they still add to 360°, you just have to remember to add the negative angles correctly.
At any given vertex, the interior angle is supplementary to an exterior angle. See Interior/Exterior angle relationship in a polygon.
In the figure above, imagine the polygon drawn on the ground. Stand on one of the sides and face along the line. Now if you walk around the polygon along each line in turn, you will eventually wind up back where you started, facing the same way. So you must have turned through a total of 360°, a full circle. This confirms that the exterior angles, taken one per vertex, add to 360°
In most geometry textbooks they say flatly that the exterior angles of a polygon add to 360° This is only true if:
So if you are asked "What is the sum of the exterior angles of a polygon?" without any conditions, you will have to guess which one they mean. Usually they mean taking one per vertex, and the answer is 360°, although strictly speaking this is wrong.