Geometry/Five Postulates of Euclidean Geometry

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Postulates in geometry are very similar to axiom of ancient Greek geometric knowledge. They are as follows:

  1. A straight line segment may be drawn from any given point to any other.
  2. A straight line may be extended to any finite length.
  3. A circle may be described with any given point as its center and any distance as its radius.
  4. All right angles are congruent.
  5. If a straight line intersects two other straight lines, and so makes the two interior angles on one side of it together less than two right angles, then the other straight lines will meet at a point if extended far enough on the side on which the angles are less than two right angles. (Proof could be a triangle because it has to be 180 degrees)

Postulate 5, the so-called Parallel Postulate was the source of much annoyance, probably even to Euclid, for being so relatively prolix. Mathematicians have a peculiar sense of aesthetics that values simplicity arising from simplicity, with the long complicated proofs, equations and calculations needed for rigorous certainty done behind the scenes, and to have such a long sentence amidst such other straightforward, intuitive statements seems awkward. As a result, many mathematicians over the centuries have tried to prove the results of the Elements without using the Parallel Postulate, but to no avail. However, in the past two centuries, assorted non-Euclidean geometries have been derived based on using the first four Euclidean postulates together with various negations of the fifth.