congruence axioms

General Congruence Relations. Let $A$ be a set and$X=A\times A$. A relation on $X$ is said to be a congruencerelation on $X$, denoted $\cong$, if the following three conditionsare 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, thecongruence relation is an equivalence relation on pairs of elementsof $A$.

Congruence Axioms in Ordered Geometry. Let $(A,B)$ be anordered geometry with strict betweenness relation $B$.We say that the ordered geometry $(A,B)$ satisfies the congruenceaxioms 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 aline ${\mathrm{\ell }}_1$ 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 aline ${\mathrm{\ell }}_2$ 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 $H$ suchthat $(a,c)\cong (p,r)$ and $(b,c)\cong (q,r)$.

Congruence Relations on line segments, triangles, andangles. With the above five congruence axioms, one may define acongruence 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 linesegment with endpoints $a$ and $b$.

It is obvious that the congruence relation defined on line segmentsof $A$ is an equivalence relation. Next, one defines a congruencerelation on triangles in $A$: $\mathrm{△}abc\cong \mathrm{△}pqr$ iftheir 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 atriangle $\mathrm{△}abc$, such that $\overline{ab}$ is congruent toa given line segment $\overline{pq}$. Then there is exactly onepoint $r$ on a chosen side of the line $\overleftrightarrow{pq}$ such that$\mathrm{△}abc\cong \mathrm{△}pqr$. Not surprisingly, the congruencerelation on triangles is also an equivalence relation.

The last major congruence relation in an ordered geometry to bedefined 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 meanthat $\mathrm{\angle }abc$ is congruent to $\mathrm{\angle }pqr$. Clearly for anypoints $x\in \overrightarrow{ba}$ and $y\in \overrightarrow{bc}$, we have $\mathrm{\angle }xby\cong \mathrm{\angle }abc$, so that $\cong$ is reflexive. $\cong$ is alsosymmetric and transitive (as the properties are inherited from thecongruence relation on triangles). Therefore, the congruencerelation on angles also defines an equivalence relation.