Congruence (geometry)

In geometry, two shapes are called congruent if one can be transformed into the other by an isometry, i.e. a combination of translations, rotations and reflections.

Note: This article is about congruences in geometry. For notions of congruence in algebra, see congruence relation.

Definition of congruence in analytic geometry

In a Euclidean system, congruence is fundamental; it's the counterpart of an equals sign in numerical analysis. In analytic geometry, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points A and B in the first mapping, it is possible to find two corresponding points α and β in the second mapping so that the Euclidean distance between A and B is equal to the corresponding Euclidean distance between α and β.

A more formal definition: two subsets A and B of Euclidean space Rn are called congruent if there exists an isometry f : Rn → Rn (an element of the Euclidean group E(n)) with f(A) = B. Congruence is an equivalence relation.

Two sets that are not congruent are called non-congruent.

Congruence of triangles

Two triangles are congruent if their corresponding sides and angles are equal. Usually it is sufficient to establish the equality of three corresponding parts and use one of the following results to conclude the congruence of the two triangles:

SAS (Side-Angle-Side): Two triangles are congruent if a pair of corresponding sides and the included angle are equal.

SSS (Side-Side-Side): Two triangles are congruent if their corresponding sides are equal.

ASA (Angle-Side-Angle): Two triangles are congruent if a pair of corresponding angles and the included side are equal.

In most system of axioms, the three criteria — SAS, SSS and ASA — are established as theorems. However, in the infamous S.M.S.G. system which heralded the short lived infatuation with the New Math stream in mathematics education, SAS is taken as one (#16) of 22 postulates.

While the AAS (Angle-Angle-Side) condition also guarantees congruence, SSA (Side-Side-Angle) does not, as there are often two dissimilar triangles with a pair of corresponding sides and a non-included angle equal. This is known as the ambiguous case. Of course, AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence shows only similarity and not congruence.

However, a special case of the SSA condition is the HL (Hypotenuse-Leg) condition. This is true because all right triangles (which this condition is used with) have a congruent angle (the right angle). If the hypotenuse and a certain leg of a triangle are congruent to the corresponding hypotenuse and leg of a different triangle, the two triangles are congruent.