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

Contents

Definition of congruence in analytic geometry

Congruence of triangles

SAS, SSS, and ASA

SSA: The ambiguous case

AAA

See also

External links

Definition of congruence in analytic geometry

In a Euclidean system, congruence is fundamental; it is 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 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 'R'^''n'' are called congruent if there exists an isometry ''f'' : 'R'^''n'' → 'R'^''n'' (an element of the Euclidean group ''E''(''n'')) with ''f''(''A'') = ''B''. Congruence is an equivalence relation.

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, SSS, and ASA

'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. However, in the infamous SMSG system which heralded the short lived infatuation with the New Math stream in mathematics education, 'SAS' is taken as one (#15) of 22 postulates.

SSA: The ambiguous case

While the 'AAS' (Angle-Angle-Side) condition also guarantees congruence, 'SSA' (Side-Side-Angle) does ''not'', because it is possible to have two incongruent triangles that satisfies the 'SSA' conditions (two congruent corresponding sides and a congruent non-included angle). This is known as the 'ambiguous case'. Specifically, 'SSA' is invalid if the non-included angle is acute.
Thus, the 'SSA' condition does prove congruence when the angle is a right angle. This is known as the 'HL' (Hypotenuse-Leg) condition, or the 'RHS' (Right Angle-Hypotenuse-Side) 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 leg of a triangle are congruent to the hypotenuse and leg of a different triangle, the two triangles are congruent.

AAA

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

See also

★ Euclidean plane isometry

★ CPCTC

External links

★ The SSS

★ The SSA

★ Congruent angles With interactive animation

★ Congruent line segments With interactive animation