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congruence axioms
General Congruence Relations. Let $A$ be a set and $X=A\times A$. A relation on $X$ is said to be a congruence relation on $X$, denoted $\cong$, if the following three conditions are satisfied:
1. $(a,b)\cong(b,a)$, for all $a,b\in A$,
2. if $(a,a)\cong(b,c)$, then $b=c$, where $a,b,c\in A$,
3. if $(a,b)\cong(c,d)$ and $(a,b)\cong(e,f)$, then $(c,d)\cong(e,f)$, for any $a,b,c,d,e,f\in A$.
By applying $(b,a)\cong(a,b)$ twice, we see that $\cong$ is reflexive according to the third condition. From this, it is easy to that $\cong$ is symmetric, since $(a,b)\cong(c,d)$ and $(a,b)\cong(a,b)$ imply $(c,d)\cong(a,b)$. Finally, $\cong$ is transitive, for if $(a,b)\cong(c,d)$ and $(c,d)\cong(e,f)$, then $(c,d)\cong(a,b)$ because $\cong$ is symmetric and so
$(a,b)\cong(e,f)$ by the third condition. Therefore, the
congruence relation is an equivalence relation on pairs of elements
of $A$.
Congruence Axioms in Ordered Geometry. Let $(A,B)$ be an
ordered geometry with strict betweenness relation $B$.
We say that the ordered geometry $(A,B)$ satisfies the congruence
axioms if
1. there is a congruence relation $\cong$ on $A\times A$;
2. if $(a,b,c)\in B$ and $(d,e,f)\in B$ with

$(a,b)\cong(d,e)$, and

$(b,c)\cong(e,f),$
then $(a,c)\cong(d,f)$;

3. 4. given the following:

three rays emanating from $p_{2}$ such that they intersect with a line $\ell_{2}$ at $a_{2},b_{2},c_{2}$ with $(a_{2},b_{2},c_{2})\in B$,

$(a_{1},b_{1})\cong(a_{2},b_{2})$ and $(b_{1},c_{1})\cong(b_{2},c_{2})$,

$(p_{1},a_{1})\cong(p_{2},a_{2})$ and $(p_{1},b_{1})\cong(p_{2},b_{2})$,
then $(p_{1},c_{1})\cong(p_{2},c_{2})$;
5. given three distinct points $a,b,c$ and two distinct points $p,q$ such that $(a,b)\cong(p,q)$. Let $H$ be a closed half plane with boundary $\overleftrightarrow{pq}$. Then there exists a unique point $r$ lying on $H$ such that $(a,c)\cong(p,r)$ and $(b,c)\cong(q,r)$.
Congruence Relations on line segments, triangles, and angles. With the above five congruence axioms, one may define a congruence relation (also denoted by $\cong$ by abuse of notation) on the set $S$ of closed line segments of $A$ by
$\overline{ab}\cong\overline{cd}\qquad\mbox{ iff }\qquad(a,b)\cong(c,d),$ 
where $\overline{ab}$ (in this entry) denotes the closed line segment with endpoints $a$ and $b$.
It is obvious that the congruence relation defined on line segments of $A$ is an equivalence relation. Next, one defines a congruence relation on triangles in $A$: $\triangle abc\cong\triangle pqr$ if their sides are congruent:
1. $\overline{ab}\cong\overline{pq}$,
2. $\overline{bc}\cong\overline{qr}$, and
3. $\overline{ca}\cong\overline{rp}$.
With this definition, Axiom 5 above can be restated as: given a
triangle $\triangle abc$, such that $\overline{ab}$ is congruent to
a given line segment $\overline{pq}$. Then there is exactly one
point $r$ on a chosen side of the line $\overleftrightarrow{pq}$ such that
$\triangle abc\cong\triangle pqr$. Not surprisingly, the congruence
relation on triangles is also an equivalence relation.
The last major congruence relation in an ordered geometry to be
defined is on angles: $\angle abc$ is congruent to $\angle pqr$ if there are
1. a point $a_{1}$ on $\overrightarrow{ba}$,
2. a point $c_{1}$ on $\overrightarrow{bc}$,
3. a point $p_{1}$ on $\overrightarrow{qp}$, and
4. a point $r_{1}$ on $\overrightarrow{qr}$
such that $\triangle a_{1}bc_{1}\cong\triangle p_{1}qr_{1}$.
It is customary to also write $\angle abc\cong\angle pqr$ to mean
that $\angle abc$ is congruent to $\angle pqr$. Clearly for any
points $x\in\overrightarrow{ba}$ and $y\in\overrightarrow{bc}$, we have $\angle xby\cong\angle abc$, so that $\cong$ is reflexive. $\cong$ is also
symmetric and transitive (as the properties are inherited from the
congruence relation on triangles). Therefore, the congruence
relation on angles also defines an equivalence relation.
References
 1 D. Hilbert, Foundations of Geometry, Open Court Publishing Co. (1971)
 2 K. Borsuk and W. Szmielew, Foundations of Geometry, NorthHolland Publishing Co. Amsterdam (1960)
 3 M. J. Greenberg, Euclidean and NonEuclidean Geometries, Development and History, W. H. Freeman and Company, San Francisco (1974)
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