geometric congruence
Two geometrical constructs are congruent if there is a finite sequence^{} of geometric transformations mapping each one into the other. In this entry, we discuss three types of geometric congruences: congruence^{} (the usual congruence), affine congruence, and projective congruence. After discussing congruence, we will briefly discuss congruence in NonEuclidean geometry before moving on to affine congruence.
Euclidean Congruence
In the usual Euclidean space, these rigid motions^{} are translations^{}, rotations, and reflections (and of course compositions of them).
In a less formal sense, saying two constructs are congruent amounts to saying that the two constructs are essentially “the same” under the geometry^{} that is being used.
The following are criteria that indicate that two given triangles^{} are congruent:

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SSS. If two triangles have their corresponding sides equal, they are congruent.

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SAS. If two triangles have two corresponding sides equal as well as the angle between them, the triangles are congruent.

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ASA. If two triangles have 2 pairs of corresponding angles equal, as well as the side between them, the triangles are congruent.

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AAS. (Also known as SAA.) If two triangles have 2 pairs of corresponding angles equal, as well as a pair of corresponding sides which is not in between them, the triangles are congruent.
Congruence in NonEuclidean Geometry
Note that the criteria listed above are also valid in hyperbolic geometry (and therefore in neutral geometry). Also note that AAS is not valid in spherical geometry, but all of the other criteria are. On the other hand, in both hyperbolic geometry and spherical geometry, AAA is a criterion that indicates that two given triangles are congruent.
Affine Congruence
Two geometric figures in an affine space^{} are affine congruent if there is an affine transformation mapping one figure to another. Since lengths and angles are not preserved by affine transformations, the class of specific geometric configurations^{} is wider than that of the class of the same geometric configuration under Euclidean congruence. For example, all triangles are affine congruent, whereas Euclidean congruent triangles are confined only to those that are SSS, SAS, or ASA. Another example is found in the class of ellipses^{}, which contains ellipses of all sizes and shapes, including circles. However, in Euclidean congruence, circles are only congruent to circles of the same radius.
Projective Congruence
Two geometric figures in a projective space are projective congruent if there is a projective transformation mapping one figure to another. Here, we find the class of congruent objects of a geometric shape even wider than the class of congruent objects of the same geometric shape under affine congruence. For example, the class of all conic sections are projectively congruent, so a circle is projectively the same as an ellipse, as a parabola^{}, and as a hyperbola^{}. Of course, under affine congruence, parabola, hyperbola, and ellipse are three distinct geometric objects.
Title  geometric congruence 
Canonical name  GeometricCongruence 
Date of creation  20130322 12:11:21 
Last modified on  20130322 12:11:21 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  22 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 51M99 
Synonym  Euclidean congruent 
Synonym  affine congruent 
Synonym  projective congruent 
Synonym  congruent 
Related topic  TriangleSolving 
Related topic  UsingPrimitiveRootsAndIndexToSolveCongruences 
Related topic  NormalOfPlane 
Related topic  CenterNormalAndCenterNormalPlaneAsLoci 
Defines  SSS 
Defines  SAS 
Defines  ASA 
Defines  SAA 
Defines  AAS 
Defines  Euclidean congruence 
Defines  affine congruence 
Defines  projective congruence 