# Thread: Interior angle sum of quadrilaterals

1. ## Interior angle sum of quadrilaterals

True or false?

In the plane, the interior angle sum of all convex and concave quadrilaterals is 360 degrees.

2. ## Re: Interior angle sum of quadrilaterals

Spoiler:
True. Quadrilateral Sum Conjecture

3. ## Re: Interior angle sum of quadrilaterals

Originally Posted by greg1313
True or false?

In the plane, the interior angle sum of all convex and concave quadrilaterals is 360 degrees.
Do you see why? Think about drawing a diagonal to make two triangles.

4. ## Re: Interior angle sum of quadrilaterals

Originally Posted by Walagaster
Do you see why? Think about drawing a diagonal to make two triangles.
It's a math puzzle. He's not asking for help.

5. ## Re: Interior angle sum of quadrilaterals

Originally Posted by greg1313
True or false?
In the plane, the interior angle sum of all convex and concave quadrilaterals is 360 degrees.
Originally Posted by romsek
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.

6. ## Re: Interior angle sum of quadrilaterals

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.

7. ## Re: Interior angle sum of quadrilaterals

Originally Posted by greg1313
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.
Going back to Hilbert's axioms of the 1890's, no angle has a measure greater that $\pi(180^o)$.
In axiomatic geometry what you are calling a reflex angle does not exist.
I carefully stated the context of my comments.

8. ## Re: Interior angle sum of quadrilaterals

Originally Posted by Plato & my edit
Oh my goodness, I continue to troll greg1313.
It is true:

In the plane, the interior angle sum of all convex and concave quadrilaterals is 360 degrees.

9. ## Re: Interior angle sum of quadrilaterals

Thread closed.

-Dan