# congruence axioms

. 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. 1.

$(a,b)\cong(b,a)$, for all $a,b\in A$,

2. 2.

if $(a,a)\cong(b,c)$, then $b=c$, where $a,b,c\in A$,

3. 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 $A$.

. 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. 1.

there is a congruence relation $\cong$ on $A\times A$;

2. 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. 3.

given $(a,b)$ and a ray $\rho$ emanating from $p$, there exists a unique point $q$ lying on $\rho$ such that $(p,q)\cong(a,b)$;

4. 4.

given the following:

• three rays emanating from $p_{1}$ such that they intersect with a line $\ell_{1}$ at $a_{1},b_{1},c_{1}$ with $(a_{1},b_{1},c_{1})\in B$, and

• 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. 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 $(b,c)\cong(q,r)$.

. 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. 1.

$\overline{ab}\cong\overline{pq}$,

2. 2.

$\overline{bc}\cong\overline{qr}$, and

3. 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. 1.

a point $a_{1}$ on $\overrightarrow{ba}$,

2. 2.

a point $c_{1}$ on $\overrightarrow{bc}$,

3. 3.

a point $p_{1}$ on $\overrightarrow{qp}$, and

4. 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, North-Holland Publishing Co. Amsterdam (1960)
• 3 M. J. Greenberg, Euclidean and Non-Euclidean Geometries, Development and History, W. H. Freeman and Company, San Francisco (1974)
Title congruence axioms CongruenceAxioms 2013-03-22 15:31:59 2013-03-22 15:31:59 CWoo (3771) CWoo (3771) 16 CWoo (3771) Axiom msc 51F20 axioms of congruence congruence relation