This Geometer's Sketchpad file contain shows a picture of a changeable polygon. By moving points A, B, C, E, or F you can change the shape of the polygon. Hide/Show buttons can be clicked on to reveal the measure of each exterior angle and the total of the exterior angles. By moving the points A, B, C, E, or F you can see the figure changes, the individual exterior angles change, but the total measure is always the same.
To give a reason why this is true, you can collapse the polygon by moving the point D toward the center until the figure collapse to 5 exterior angles only. You should see a complete 360 degrees.
A second page allows you to repeat the same concept using a different polygon. Pull on point B to change the size of the polygon.