Why the Polygon Interior Angle Sum Theorem does not apply for concave polygon?

Question

Why the Polygon Interior Angle Sum Theorem does not apply for concave polygon?

 

Apparently this theorem should hold true for simple concave polygon, but your website states that "As seen in the statement above, the polygon must be convex, which is not a term we have studied yet. What this means is just that the polygon cannot have angles that point in. If a polygon does have an angle that points in, it is called concave, and this theorem does not apply. In other words, all of the interior angles of the polygon must have a measure of no more than 180° for this theorem to work."

Answer 1

Whose website?

"

The interior angles of any polygon always add up to a constant value, which depends only on the number of sides. For example the interior angles of a pentagon always add up to 540° no matter if it is regular or irregular, convex or concave, or what size and shape it is. The sum of the interior angles of a polygon is given by the formul."

Answer 2

This answer was really intended as an answer to the comment on Robert's answer.

 

I believe the wyzant reference quoted in the comment is incorrect.  To see this, let's first take the sample pentagon at at http://www.wyzant.com/resources/lessons/math/geometry/quadrilaterals/polygons .

 

Draw lines from the vertex where the angle is greater than 180 degrees to the other two opposing vertices forming three triangles.  Each of the 3 triangles has angles totaling 180 degrees.  

 

Now assert that the sum of the angles in all three triangles add up to  the sum of the interior angles of the pentagon.  You can see this visually.   The sum of all the angles in the 3 triangles is 3*180 degrees which is indeed (n-2)*180 where n = 5.

 

Exterior angles are the ones affected by the polygon being irregular, not interior angles.  In an irregular polygon the sum of the exterior angles is not 360 degrees unless you change the rule to subtracting exterior angles which are inside the polygon, as shown at http://www.mathopenref.com/polygonanglerelation.html.