congruence axioms

General Congruence RelationsPlanetmathPlanetmathPlanetmath. Let A be a set and X=A×A. A relation on X is said to be a congruence relation on X, denoted , if the following three conditions are satisfied:

  1. 1.

    (a,b)(b,a), for all a,bA,

  2. 2.

    if (a,a)(b,c), then b=c, where a,b,cA,

  3. 3.

    if (a,b)(c,d) and (a,b)(e,f), then (c,d)(e,f), for any a,b,c,d,e,fA.

By applying (b,a)(a,b) twice, we see that is reflexiveMathworldPlanetmathPlanetmath according to the third condition. From this, it is easy to that is symmetricMathworldPlanetmathPlanetmath, since (a,b)(c,d) and (a,b)(a,b) imply (c,d)(a,b). Finally, is transitiveMathworldPlanetmathPlanetmathPlanetmath, for if (a,b)(c,d) and (c,d)(e,f), then (c,d)(a,b) because is symmetric and so (a,b)(e,f) by the third condition. Therefore, the congruence relation is an equivalence relationMathworldPlanetmath on pairs of elements of A.

Congruence Axioms in Ordered GeometryMathworldPlanetmath. 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 on A×A;

  2. 2.

    if (a,b,c)B and (d,e,f)B with

    • (a,b)(d,e), and

    • (b,c)(e,f),

    then (a,c)(d,f);

  3. 3.

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

  4. 4.

    given the following:

    • three rays emanating from p1 such that they intersect with a line 1 at a1,b1,c1 with (a1,b1,c1)B, and

    • three rays emanating from p2 such that they intersect with a line 2 at a2,b2,c2 with (a2,b2,c2)B,

    • (a1,b1)(a2,b2) and (b1,c1)(b2,c2),

    • (p1,a1)(p2,a2) and (p1,b1)(p2,b2),

    then (p1,c1)(p2,c2);

  5. 5.

    given three distinct points a,b,c and two distinct points p,q such that (a,b)(p,q). Let H be a closed half plane with boundary pq. Then there exists a unique point r lying on H such that (a,c)(p,r) and (b,c)(q,r).

Congruence Relations on line segmentsMathworldPlanetmath, trianglesMathworldPlanetmath, and angles. With the above five congruence axioms, one may define a congruence relation (also denoted by by abuse of notation) on the set S of closed line segments of A by

ab¯cd¯   iff   (a,b)(c,d),

where ab¯ (in this entry) denotes the closed line segment with endpointsMathworldPlanetmath 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: abcpqr if their sides are congruent:

  1. 1.


  2. 2.

    bc¯qr¯, and

  3. 3.


With this definition, Axiom 5 above can be restated as: given a triangle abc, such that ab¯ is congruent to a given line segment pq¯. Then there is exactly one point r on a chosen side of the line pq such that abcpqr. 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: abc is congruent to pqr if there are

  1. 1.

    a point a1 on ba,

  2. 2.

    a point c1 on bc,

  3. 3.

    a point p1 on qp, and

  4. 4.

    a point r1 on qr

such that a1bc1p1qr1.

It is customary to also write abcpqr to mean that abc is congruent to pqr. Clearly for any points xba and ybc, we have xbyabc, so that is reflexive. 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.


  • 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
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