Definition: Congruence

Two subsets \(X\) and \(Y\) of Euclidean metric space \(\mathbb R^n\) are called congruent, if there exist an Euclidean movement (or a composition of such movements), that reversibly transforms every point of \(X\) into a point of \(Y\).

Special Case Euclidean 1-dimensional Space (Straight Line) \(\mathbb R^1\)

We call two segments on a straight line congruent, if they are identical in size and position, or if we can find one or more than one Euclidean movements (like translations and/or reflections) such that both segments become identical in size and position.

Special Case Euclidean 2-dimensional Space (Plane) \(\mathbb R^2\)

We call two plane figures (e.g. triangles, rectangles, trapezoids, circles) in a plane congruent, if they are identical in size and position, or if we can find one or more than one Euclidean movements (like translations, reflections and/or rotations) such that both figures become identical in size and position.

Special Case Euclidean 3-dimensional Space \(\mathbb R^3\)

We call two solids (e.g. polyhedrons, spheres) congruent, if they are identical in size and position, or if we can find one or more than one Euclidean movements (like translations, reflections and/or rotations) such that both solids become identical in size and position.

Axioms: 1
Definitions: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Explanations: 18
Proofs: 19 20 21 22 23 24 25
Propositions: 26 27 28 29 30 31 32 33 34 35 36 37 38 39


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References

Bibliography

  1. Knerr, Richard: "Knaurs Buch der Mathematik", Droemer Knaur Lexikographisches Institut, M√ľnchen, 1989

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"