Congruence (geometry)

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An example of congruence. The two figures on the left are congruent, while the third is similar to them. The last figure is neither. Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles. The latter sort of properties are called invariants and studying them is the essence of geometry.
An example of congruence. The two figures on the left are congruent, while the third is similar to them. The last figure is neither. Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles. The latter sort of properties are called invariants and studying them is the essence of geometry.

In geometry, two sets of points are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Less formally, two figures are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply placed somewhere else).

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[edit] Definition of congruence in analytic geometry

In a Euclidean system, congruence is fundamental; it is the counterpart of equality for numbers. 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 in the first mapping, the Euclidean distance between them is equal to the Euclidean distance between the corresponding points in the second mapping.

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.

[edit] 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.

The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and an adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, usually yields two distinct possible triangles.
The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and an adjacent side (AAS). Specifying two sides and an adjacent angle (SSA), however, usually yields two distinct possible triangles.

[edit] SAS, SSS, ASA, and AAS

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. The ASA Postulate was contributed by Thales of Miletus (Greek).

In most system of axioms, the three criteria — SAS, SSS and ASA — are established as theorems. In the School Mathematics Study Group system SAS is taken as one (#15) of 22 postulates.

AAS (Angle-Angle-Side): Two triangles are congruent if a pair of corresponding angles and a not-included side are equal, since the 3rd angle would have to be equal, and therefore the side would be included. This one is valid only in Euclidean geometry.

[edit] SSA: The potentially ambiguous case

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS (Angle-Side-Side)) does not always prove congruence.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter or equal to the adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases, SSA proves congruence. Notice that the opposite side cannot be smaller than the adjacent side times the sine of the angle as this could not describe a triangle.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition), we can calculate the third side and fall back on SSS.

The SSA condition proves congruence if the angle is acute and the opposite side either equals the adjacent side times the sine of the angle (right triangle) or is longer than the adjacent side.

[edit] AAA

AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence shows only similarity and not congruence. However, in spherical geometry and hyperbolic geometry this is sufficient for congruence.

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