True or false?
In the plane, the interior angle sum of all convex and concave quadrilaterals is 360 degrees.
I have avoided commenting but here goes. If one follows the link in romsek's quote, s/he will see "The precise statement of the conjecture is:
Conjecture (Quadrilateral Sum ): The sum of the measures of the interior angles in any convex quadrilateral is 360 degrees."
Now this question has a checked history. Anyone who has done work is axiomatic geometry understands that definitions depend upon the user.
In the diagram, the quadrilateral is non-convex. But there is disagreement on the angles of the figure. There is agreement that to adjacent sides (sides with a common vertex) determine an angle of the figure. Thus by that definition, $\angle ABC$ is an angle of the quadrilateral.
But is $\angle ABC$ part of the sum? Well according to this proof it is not. Now Alan P, that author, would not have passed anyone of my axiomatic geometry courses: $0\le m(\angle ABC)\le\pi(180^o)$. The statement that an angle of a quadrilateral is interior means that the angle union its interior contains the quadrilateral. That is clearly not the case for $\angle ABC$.
So I argue that based upon the Moore axioms of Geometry the answer to the original post is: FALSE. But definitions differ.
In the diagram in post # 5, of the four interior angles, three of them
are acute angles, and one of them is a reflex angle (greater than
180 degrees but less than 360 degrees). I don't know of any
different definition of a reflex angle. We are just concerned about
the sum of the measures of the four interior angles. If someone
can come up with a counterexample or a contradiction for concave
quadrilaterals, for example, then it is true only for the convex
quadrilaterals.