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Moise's Axioms

Axioms (Neutral Geometry) The following axioms produce neutral geometry:

Incidence Axioms:

(I-1) Each two distinct points determines a unique line.

(I-2) Three noncollinear points determines a unique plane.

(I-3) If two points lie on a plane, then any line containing those two points lies in that plane.

(I-4) If two distinct planes meet their intersect is a line.

(I-5) Space consists of at least four noncoplanar points, and contains three noncollinear points. Each plane is a set of points of which three are noncollinear, and each line is a set of at least two distinct points.

The Distance Axioms:

(D-1) Each pair of points and is associated with a unique real number, called the distance from to denoted by

(D-2) For all points and with equality only when

(D-3) For all points and

(D-4) The points of each line may be assigned to the entire set of real numbers called coordinates, in such a manner that

    (i) each point on is assigned to a unique coordinate
    
    (ii) no two points are assigned to the same coordinate
    
    (iii) any two points on may be assigned the coordinates zero and a positive real number, respectively.
    
     (iv) if points and on have coordinates and then

The Angle Axioms:

(A-1) Each angle is associated with a unique real number between 0 and 180, called its measure and denoted No angle can have measure 0 nor 180.

(A-2) If lies in the interior of then Conversely, if then is an interior point of

(A-3) The set of rays lying on one side of a given line including ray may be assigned to the entire set of real numbers called coordinates, in such a manner that

    (i) each ray is assigned to a unique coordinate
    
    (ii) no two rays are assigned to the same coordinate
    
    (iii) the coordinate of is 0
    
     (iv) if rays and on have coordinates and then
     
(A-4) A linear pair of angles is supplementary pair.

The Half-Plane Axiom:
     
(H-1) Let be any line lying in any plane The set of all points in not on consists of the union of two subsets and of such that

     (i) and are convex sets
     
     (ii) and have no points in common
     
     (iii) If lies in and lies in the line intersects the segment

The Congruence Axiom:

(C-1) If two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of another, the triangles are congruent.

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