Congruence I Propositions

    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.

Definition (Congruent Triangles) Triangles and are congruent triangles if a one-to-one correspondence can be set up between their vertices so that corresponding sides are congruent and corresponding angles are congruent. Using the convention that corresponds to to and to    and   are congruent is denoted by  

    The Side Angle Side (SAS) Axiom provides a connection between the congruence axioms which are relational for segments and those which are relational for angles. The Segment Shift Axiom says that you can "shift" the segment so that it lies on the ray with superimposed on and superimposed on The Segment Congruence Axiom and the Angle Congruence Axiom asserts the transitivity and reflexivity of the congruence relations of segment congruence and angle congruence, respectively. Further, combining together transitivity and reflexivity, it follows that both relations are symmetric: if then and if then respectively.

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

    Proof. By the Angle Shift Axiom there is a unique ray on a given side of the line such that and can be chosen to be the unique point such that by the Segment Shift Axiom. By the SAS Axiom  

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

    Proof. We need to show that if in we have then Consider the correspondence of vertices Under this correspondence and the Angle Congruence Axiom, two sides and the included angle of are congruent respectively to the corresponding sides and included angle of Namely, and and therefore by the SAS Axiom. Therefore, by the Congruent Triangles Definition.  

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
    
        Proof. (i): Assume that is not congruent to By the Segement Shift Axiom, there is a point on such that The points and are distinct because otherwise contrary to our assumption. Since and by the Additive Axiom By hypothesis and by the Segment Congruence Axiom By the Line Segment Definition By the Law Of The Excluded Middle, must be congruent to
    (ii): There is a unique point on such that by the Segment Shift Axiom. Assume for a contradiction that were not between and By the Ray Definition, either or If then and are two distinct points on such that contradicting the uniqueness part of the Segment Shift Axiom. If then there is a point on the ray opposite to such that by the Segment Shift Axiom. By the Additive Axiom, Thus, there are two distinct points and on such that contradicting the uniqueness part of the Segment Shift Axiom. Whence,



    

Definition (Segment Relation) Given points and (or ) means that there exists a point such that and

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
    
    Proof. (i): Suppose is not congruent to By the Segement Shift Axiom there exists a unique point on such that By the Order Axiom and the Ray Definition, exactly one of the following holds: or Suppose holds then and implies by the Segment Relation Definition. Suppose holds. By the Segment Subtraction Proposition, there exists a point such that and Then and implies by the Segment Relation Definition. Therefore, exactly one of the following must hold:   or
    (ii): By definition of there exists a point such that and By the Segment Subtraction with and there is a unqiue point such that and By the Segment Relation Definition,
    (iii): By the Segment Relation Definition applied to there exists a point such that and By the Segment Relation Definition,
    (iv): By the Segment Relation Definition applied to there exists a point such that and By the Segment Relation Definition applied to there exists a point such that and By the Segment Subtraction Proposition with and there exists a unqiue such that and By the Betweenness Property with and it follows and by the Segment Relation Definition,

Definition (Vertical Angles) Vertical angles are angles that admit labelings and such that and are opposite rays, and and are opposite rays.

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.
    
        Proof. (i): Given and with and supplementary angles, and supplementary angles, and it will be shown that By the Segment Shift Axiom and the Angle Shift Axiom, we can choose the points and on the sides of the other angle and its supplements so that and Then by the SAS Axiom, By the Congruent Triangles Definition, and By the Additive Axiom, By  SAS Axiom, By the Congruent Triangles Definition, and By the SAS Axiom, By the Congruent Triangles Definition, as needed.
    (ii): Given vertical angles and it will be shown that By the Vertical Angles Definition and the Supplementary Angles Definition, and are supplementary angles. By the Vertical Angles Definition and the Supplementary Angles Definition, and are supplementary angles. By the Angle Congruence Axiom, Therefore,  by part (i),
    (iii): Let be congruent to a right angle with and the angles and are supplementary angles. It will be shown that By part (i), By the Angle Congruence Axiom, By the Right Angle Definition, is a right angle.

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

    Proof. Assume first that does not lie on and let and be any two points on by the Points On Line Axiom. On the opposite side of from there exists a ray such that by the Angle Shift Axiom. There is point on such that by the Segment Shift Axiom. By the Side Of A Line Definition, intersects in a point If then by the Perpendicular Lines Definition. If then by the SAS Axiom. Hence, by the Congruent Triangles Definition, so by the Perpendicular Lines Definition. Assume now that lies on By the Points Not On Line Proposition, we can drop a perpendicular from one of them to thereby obtaining a right angle. We can layoff an angle congruent to this right angle with vertex and one side on by the Angle Shift Axiom. The other side of this angle is part of a line through perpendicular to by the  Special Angles Property Proposition.  

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

    Proof. By the Segment Shift Axiom there is a unqiue point on such that By the SAS Axiom, By the Congruent Triangles Definition, By the Angle Congruence Axiom, and implies Therefore, by the Angle Definition, By the Ray Definition, whence,   

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

    Proof. By the Segment Congruence Axiom, and by hypothesis and By the ASA Proposition, and thus, by the Congruent Triangles Definition,   

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

    Proof. By the Crossbar Proposition, we may assume that is chosen so that By the Segment Shift Axiom, we assume and chosen so that abd Then and by the Side Angle Side Proposition. By the Congruent Triangles Definition, and is supplementary to By the Angle Criterion For Congruence Proposition and the Additive Axiom, and are collinear and is supplementary to By the Interior Angle Property, By the Additive Axiom, Then By the SAS Axiom,   By the Congruent Triangles Definition,

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

    Proof. By the Angle Shift Axiom there is a unique ray on the opposite side of from such that By the Angle Addition Proposition, By Uniqueness

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
    
        Proof. (i): Let and be defined by and By the Angle Shift Axiom, there is a unique ray on the same side of as such that By the Half-Planes Proposition, exactly one of the following cases hold: is on is on the same side of as or is on the opposite side of as If is on then by the Linear Decomposition Proposition and so Therefore,  


For case 2, if is on the same side of as then is in the interior of and so by definition is between and this with implies by the Angle Relation Definition.


For case 3, suppose is on the opposite side of as Choose on on such that and Then by the SAS Axiom. Hence, Since is on the opposite side of as intersects in a point We have and By the Segment Subtraction Proposition, there is a unqiue point such that and By the SAS Axiom, So by the Congruent Triangles Definition, and thus by the Angle Subtraction Proposition, Since is between and by the Interior Angle Property. Thus by the Angle Relation Definition.  


    (ii): Let and If then there exists a point such that by the Angle Relation Definition, and such that by the Interior Angle Property. By the Segment Shift Axiom there exists points and such that and with Given that    follows by the SAS Congruence Criterion. Therefore, by the Congruent Triangles Definition and By the Segment Subtraction Proposition the exists a such that and By the SAS Criterion, By the Angle Subtraction Proposition,
    (iii): Let If then by the Angle Relation Definition there exists a in the interior of such that By the Angle Congruence Axiom Therefore, by the Angle Relation Definition.
    

    (iv):  Let be defined by points If then by the Angle Relation Definition and the Interior Angle Property there exists a point such that and Since part (ii) implies Then by the Angle Relation Definition and the Interior Angle Property there exists a point such that and The Betweenness Property with and implies and thus by the Angle Relation Definition and Interior Angle Property,   

Proposition (SSS Criterion for Congruence) Given and   If and then     

    Proof. By the Laying Off Proposition, the Segment Conguence Axiom and the Segment Shift Axiom, let and and are on opposite sides of By the Isosceles Triangle Definition, is an isosceles triangle, and therefore by the Pappus Property. By the Side Of A Line Definition, intersects in a point By the Ordering Axiom, exactly one of the following occurs: or
    Case 1: Suppose By the Linear Decomposition Proposition, and By the SAS Axiom,
    Case 2: Suppose Then, by the Interior Angle Property, is between and and also is between and By the Linear Decomposition Proposition, and By the Pappus Property, and   By the Angle Addition Proposition, we have By the SAS Axiom,
    Case 3:  Suppose By the Pappus Property, and Moreover, is between and and is between and By the Angle Subtraction Proposition, By the SAS Axiom,   
    
        

Proposition (Fourth Postulate Of Euclid) All right triangle are congruent to each other.

    Proof.  Given two right angles and By the Right Angle Definition, and Assume for a contradiction that is not congruent to By the Angle Ordering Proposition, one of these angles is smaller, say By the Angle Relation Definition, there is a ray between and such that By the Special Angle Proposition, By the Angle Congruence Axiom, By Angle Ordering Proposition, there is a ray between and such that By the Angle Congruence Axiom, and By the Angle Relation Definition, and by the  Interior Angle Property,   By the Angle Ordering Proposition, we have a contradiction to the assumption   Therefore,

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