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Geometry - question about the proceess of proving a theorem

  1. Dec 24, 2013 #1

    Please refer to the Pythagorean proof of the theorem that the angels in a triangle add to 180 degrees. The following link has the proof.

    You will note that this proof is based on the assumptions that angles on a straight line add to 180 degrees. But we have no proof of that fact. I mean we can measure it with a protractor, but is there a way to prove that?

    So I am wondering about the whole process of proving a theorem. We use some assumptions to prove a statement. The assumptions themselves are theorems that have to proved somewhere else. Is that the right way to think about it?

  2. jcsd
  3. Dec 24, 2013 #2


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    For your particular question. A complete circle is 360 deg. The angles on one side of a straight line must add up to a semicircle (sincle both sides of the line must have the same angle sum).
  4. Dec 24, 2013 #3


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    In any formal system, there are one or more assumptions which are assumed to be true but which cannot otherwise be proven. In geometry and arithmetic, these unprovable assumptions are called postulates or axioms.
    Euclidean geometry has 5 axioms and 5 other 'common notions' which are similar to axioms.


    The most controversial axiom in Euclidean geometry is the Fifth, or Parallel, postulate. Other, non-euclidean geometries have been constructed by discarding this postulate.

    In the 1930's, the logician Kurt Godel proved that in logical systems like arithmetic and geometry, there are certain axioms which are true but which cannot be proven true using only the elements contained in that system.


    In this attachment, there is a proof on pp. 20-21 that the angles in a triangle add to 180 degrees:


    which, I believe, is similar to the one you linked to.

  5. Dec 24, 2013 #4
    Just for the sake argument, if someone says to me : how can we assume that "a complete circle is 360 deg"? Can I say that, "it a system that we humans have designed (i.e. dividing a circle in 360 equal parts), so there is no need of a proof".
  6. Dec 24, 2013 #5
    The fourth postulate of Euclid is "That all right angles are equal to one another."

    1.XIII says "If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles."


    Draw a semi circle. This figure has two right angles in it (because of the straight line, the diameter). Complete the circle, and the angles double, so four right angles. Convention is one right angle is 90 degrees, so a circle has 360 degrees.

    That's not a rigorous proof, but you get the idea.
  7. Dec 25, 2013 #6


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    What is your definition of an angle?
  8. Dec 25, 2013 #7


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    This is a definition, as far as I'm concerned. Some humans long ago, the Babylonians I believe, decided to divide a circle in 360 equal parts. They could have decided to divide it into a different number of parts - the grad system divides a circle into 400 equal parts so that a quarter of the circle (a right angle) consists of 100 grads.
  9. Dec 25, 2013 #8


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    As others have said, the basic assumption here is that the angles on a straight line always add up to the same amount. Whether you call it 180 degrees, 2 right angles, pi radians, or 200 grads isn't very important.

    More interesting is the fact that the theorem is really about the properties of a plane. If you imagined that a "plane" was the curved surface of a sphere, the theorem isn't true. You can easily draw an "equilateral triangle" on a sphere where each of the 3 angles is 90 degrees. In fact the sum of the angles of a "spherical triangle" is related to the area of the triangle.

    Euclid's version of the theorem is only true if Euclid's postulate (assumption) about parallel lines is true, and that is the key assumption that distinguishes Euclidiean from non-Euclidean geometry.

    The fact that for people living on earth, triangles drawn on pieces of paper seem to be (approximately) consistent with the assumptions of Euclidean geometry, is a statement about physics, not about geometry.
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