Congruence I Proposition List

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.

Proposition (Laying Off) Given and segment there is a unique point on a given side of the line such that

Proposition (Pappus Property) Base angles are congruent in an isoceles triangle.

Proposition (Segment Subtraction) Given points and
(i) if and then and
(ii) if then for any point between and there is a unqiue point between and such that

Proposition (Segment Ordering) Given points and
    (i) exactly one of the following conditions holds: or
    (ii) if and then
    (iii) if and then and
    (iv) if and then

Proposition (Special Angles) The following hold:
    (i) supplements of congruent angles are congruent,
    (ii) vertical angles are congruent to each other, and
    (iii) any angle congruent to a right angle is a right angle.

Proposition (Point-Line Perpendicular Property) For every line and every point there exists a line through perpendicular to

Proposition (ASA Criterion for Congruence) Given and with and Then

Proposition (Isosceles Criterion) Given if then and is an isosceles triangle.

Proposition (Angle Addition) Given between and between and and Then

Proposition (Angle Subtraction) Given between and between and and Then

Definition (Angle Relation) Given two angles and means there is a ray between and such that

Proposition (Angle Ordering)
    (i) Exactly one of the following conditions hold: or
    (ii) If and then
    (iii) If and then
    (iv) If and then