be given in the plane α and let a line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one ray k′ such that the angle , or , is congruent to the angle and at the same time all interior points of the angle lie upon the given side of a′. We express this relation by means of the notation . If the angle is congruent to the angle and to the angle , then the angle is congruent to the angle ; that is to say, if and , then . If, in the two triangles ABC and A′B′C′ the congruences , , hold, then the congruence holds (and, by a change of notation, it follows that also holds). IV. Parallels Euclid's Axiom: Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and does not intersect a. V. Continuity Axiom of Archimedes: If AB and CD are any segments then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B. Axiom of line completeness: An extension (An extended line from a line that already exists, usually used in geometry) of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible. Hilbert's discarded axiom Hilbert (1899) included a 21st axiom that read as follows: II.4. Any four points A, B, C, D of a line can always be labeled so that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D. E.H. Moore and R.L. Moore independently proved that this axiom is redundant, and the former published this result in an article appearing in the Transactions of the American Mathematical Society in 1902. Before