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Finally, in the fifth part of the problem I showed that on the plane or on a hyperbolic plane, two geodesics either coincide or are disjoint or they intersect in one point. On a sphere, two geodesics either coincide or intersect exactly twice. I have provided the homework assignment as an artifact so that one can read through my explanations. Instead of rewriting them for this part of the essay I decided to just include my previous work. I also discovered why the Isosceles Triangle Theorem (ITT) held on all three of the surfaces. I have provided the homework assignment as an artifact for this because I would need to explain the entire proof. So, this gives the reader the opportunity to see why the Isosceles Triangle Theorem holds on all three surfaces. For this homework assignment I also proved the Corollary for the ITT and the converse of the ITT. One can see these by looking at my homework assignment Through this class we have also looked at many triangle congruencies. The first one that we looked at was Side-Angle-Side. First we proved this congruency on a plane. Then we took the idea to a sphere. However, when we took the idea to a sphere we noticed that we had to restrict our triangle. So, I was given an assignment that defined different types of small triangles. I was to explain if the definition allowed Side-Angle-Side congruency to hold. I have included this