Euclidean Geometry

A.

Neutral geometry is also known as absolute geometry. Euclidean Geometry uses all five of Euclid’s postulates whereas; Neutral Geometry is the study of geometry without the fifth postulate. In Euclidean geometry, the unique parallel line is very important. Without the unique parallel line; however, neutral geometry is very limited. Neutral geometry is consistent with hyperbolic geometry and Euclidean geometry, but not with elliptic geometry. In absolute geometry, the parallel theorem reads: “If two lines in the same plane are cut by a transversal so that a pair of alternate interior angles are congruent, the lines are parallel” (Kay, 2000, p. 213). However, the converse of this is Euclidean parallel lines postulate which reads: “If two lines in the same place are cut by a transversal so that the sum of the measures of a pair of interior angles on the same side of the transversal is less than 180, the lines will meet on that side of the transversal” (Kay, 2000, p. 213).

B.

Euclid’s parallel postulate tells us that if we have a line n and a point not on that line, point D, there is exactly one line parallel to line n that goes through point D. Even Euclid was not comfortable using this axiom. Mathematicians struggled to show that it could be proven using the first 4 axioms. Even Euclid didn’t feel comfortable using the fifth axiom. They then divided the geometry into two sections: basic geometry and Euclidean geometry. Basic geometry was based on the first four axioms alone and Euclidean geometry used all 5 axioms (Appendix, p. 1). Mathematicians tried to prove that the 5th axiom could be proven using only the first four axioms directly. They then tried proving using the first four axioms indirectly. Two guys, Girolamo Saccheri and Johann Lambert, worked with non-Euclidean geometry to try and prove the 5th axiom. Saccheri tried using a contradiction to the fifth axiom. He worked with a quadrilateral and found that the angles were either right, obtuse or...