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Hilbert's Congruence Axioms

    A purpose of the Hilbert Congruence Axioms is to give meaning to the undefined term congruence; seeing as congruence is undefined, we have only the axioms to guarantee that the points in this geometry behave in a way that is consistent with our interpretation of "congruence". This topic points out that the Hilbert Congruence Axioms do give segment and angle congruence as congruence relations. Addition and subtraction of segments and angles are detailed. More relations are defined for angles and segments and trichotomy properties are detailed. The ASA Congruence Criterion and Isosceles Criterion are proven.

Axiom (Congruence Axioms) The following axioms are called the Congrunce Axioms.

    (i) (Segment Shift) If and are distinct points and if is any point, then for each ray emanating from there is a unique point on such that and
    
    (ii) (Segment Congruence) If and then Moreover, every segment is congruent to itself.
    
    (iii) (Additive) If , and then
    
    (iv) (Angle Shift) Given and there is a unique ray on a given side of such that
    
    (v) (Angle Congruence) If and then Moreover, every angle is congruent to itself.
    
    (vi) (Side Angle Side) If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.

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