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Thread: Interior angle sum of quadrilaterals

  1. #1
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    Interior angle sum of quadrilaterals

    True or false?

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

    Spoiler:
    True. Quadrilateral Sum Conjecture
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    Re: Interior angle sum of quadrilaterals

    Quote Originally Posted by greg1313 View Post
    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.
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    Re: Interior angle sum of quadrilaterals

    Quote Originally Posted by Walagaster View Post
    Do you see why? Think about drawing a diagonal to make two triangles.
    It's a math puzzle. He's not asking for help.
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    Re: Interior angle sum of quadrilaterals

    Quote Originally Posted by greg1313 View Post
    True or false?
    In the plane, the interior angle sum of all convex and concave quadrilaterals is 360 degrees.
    Quote Originally Posted by romsek View Post
    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.Interior angle sum of quadrilaterals-ncvq.gif
    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.
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    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.
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    Re: Interior angle sum of quadrilaterals

    Quote Originally Posted by greg1313 View Post
    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.
    Last edited by topsquark; Oct 13th 2018 at 10:15 PM. Reason: Argumentative
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    Re: Interior angle sum of quadrilaterals

    Quote Originally Posted by Plato & my edit View Post
    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.
    Last edited by greg1313; Oct 13th 2018 at 08:02 AM.
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    Forum Admin topsquark's Avatar
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    Re: Interior angle sum of quadrilaterals

    Thread closed.

    -Dan
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