In my previous post I have already mentioned several axioms things that refer to the flatness of a plane. But there are others that are less obvious. The most famous is known as Euclid's 5th Axiom also known as the parallel axiom.

Euclid's 5th Axiom states: "If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines if produced indefinitely meet on that side which are the angles less than two right angles." This is basically the contrapositive (basically a rewording) of Holt's Same-Side Interior Angle Theorem  which states, "If two parallel lines are crossed by a transversal [a straight line crossing the two], then the same-side interior angles are supplementary [their measures add up to 180 degrees i.e. the equivalent of two right angles]."

Of course, theorems are statements which are proven, i.e. demonstrated logically as a consequence of definitions and axioms. The axiom Holt uses to prove the same-side interior angle theorem is called the Corresponding Angles Axiom: "If two parallel lines are crossed by a transversal, then corresponding angles are congruent[have the same measure]." This statement may be a little different, but is logically equivalent, meaning that if either one is assumed, then the other can be proven. Interestingly, what Holt labels as its parallel axiom is not the Corresponding Angles Theorem. Holt's Parallel Axiom tells us that given a line and a point not on the line, that there exists exactly one line parallel to the given line passing through the point. This is used as a "more rigorous" version of Euclid's parallel axiom by Hilbert, who gave Playfair credit for the suggestion, who gave credit to someone else.

Euclid's 5th axiom is famous because of the many attempts that had been made to prove it on the basis of the other axioms and because of its role in modern geometry. Since the axiom seems more complex and perhaps even less obvious than others, for hundreds of years many geometers thought to improve upon Euclid by proving it. All attempts seemed to fail, until someone decided to throw out the axiom all together and create an alternative geometry. It was found that an alternative geometry that is consistent could exist, albeit some of the conclusions would necessarily change or become more generalized. These geometries made by omitting Euclid's 5th axiom are referred to as modern geometries. It was also discovered that the validity of the axiom was dependent upon the flatness of the surface. Thus modern geometries are the geometries of the smooth curved surfaces, not flat.

Interestingly, once the revelation regarding the association of Euclid's 5th axiom and flatness was uncovered it was discovered that many things proven using Euclid's 5th axiom were logically equivalent and could be substituted. Among the many equivalent statements, the most interesting to me are the Pythagorean Theorem and the Triangle Sum Theorem. The Pythagorean Theorem states that the sum of the squares of the lengths of the legs of a right triangles is equal to square of the length of the hypotenuse. The Triangle Sum Theorem states that the sum of the interior angles of a triangle is 180 degrees.As a result the Pythagorean Theorem has been used to try to determine whether space itself is flat/straight versus curved, by measuring angles and distances between distant objects. Einstein's Theory of Relativity implies the possibility that space is not necessarily "Euclidean", i.e. flat in a three dimensional sense.

As for the triangle sum theorem, the sum of the angles of a triangle on a convex surface is greater than 180 degrees and less than 180 degrees on a concave surface. Furthermore, on a flat surface the Triangle Sum Theorem implies that a triangle can have at most one right angle. However, on a sphere it can have three.
 To draw such a triangle on a ball or balloon, think of it as a globe. Begin at the north pole and draw a segment downward towards the south pole, but stop at the equator. The next side should continue along the equator one-quarter of the way around, then the last side will end back at the north pole where you started.

Interestingly, many of the definitions that we use for flat surfaces can still be applied to curved surfaces. We can still think of a line as the extension of the shortest path between two points, on a given surface. This added stipulation in relation to the surface allows me to maintain other axioms as well such as, "if two points lie on the surface then the line containing those two points is also on the surface." Although the line would not be straight in the way it would be on a plane, it still would be the extension of the most direct route between two points.  This is the type of consideration that airline pilots must make when flying around the globe. The straight path on a flat map, is slightly different that the direct route on a round globe.