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. 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. given the following:

   • three rays emanating from $p_1$ such that they intersect with a line ${\mathrm{\ell }}_1$ 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 ${\mathrm{\ell }}_2$ 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

$$\overline{ab}\cong \overline{cd}\mathit{\hspace{1em}\hspace{1em}}\text{ iff }\mathit{\hspace{1em}\hspace{1em}}(a,b)\cong (c,d),$$

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 $A$: $\mathrm{△}abc\cong \mathrm{△}pqr$ 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 $\mathrm{△}abc$, 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 $\mathrm{△}abc\cong \mathrm{△}pqr$. 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: $\mathrm{\angle }abc$ is congruent to $\mathrm{\angle }pqr$ 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 $\mathrm{△}{a}_{1}b{c}_1\cong \mathrm{△}{p}_{1}q{r}_1$.

It is customary to also write $\mathrm{\angle }abc\cong \mathrm{\angle }pqr$ to mean that $\mathrm{\angle }abc$ is congruent to $\mathrm{\angle }pqr$. Clearly for any points $x\in \overrightarrow{ba}$ and $y\in \overrightarrow{bc}$, we have $\mathrm{\angle }xby\cong \mathrm{\angle }abc$, 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.

Title	congruence axioms
Canonical name	CongruenceAxioms
Date of creation	2013-03-22 15:31:59
Last modified on	2013-03-22 15:31:59
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	16
Author	CWoo (3771)
Entry type	Axiom
Classification	msc 51F20
Synonym	axioms of congruence
Defines	congruence relation