Congruence (geometry)

In geometry, two figures are congruent ( identical or uniform) if they can be converted into each other by a congruence (from Latin congruens = consistent, appropriate). Kongruenzabbildungen (also called movements ) are parallel translation, rotation, reflection, and the links of these images.

The congruence of two plane geometric figures can be clearly so indicate: You can cut out a figure with scissors and then place it on the other, that both are superimposed exactly each other so exactly " cover " (→ compare congruence ). This is called congruent plane figures therefore congruent. Figures, which are not to be congruent, also called incongruent.

Congruent planar polygons and polyhedra spatial distinguished in that respective track lengths and angle sizes match.

In absolute geometry two figures are called congruent, if a movement of the point of space exists through which the one character is mapped bijectively on the other.

Example

The first two characters are congruent. While the third party has the same shape, but is smaller. Therefore, it is similar to the first and second figure, but not congruent. The last figure is not the same shape, and is therefore neither similar nor congruent with the T-shaped figures.

Congruence of triangles

Especially easy is the congruence of triangles checked using the following five congruence theorems that provide simple criteria under which two triangles are congruent:

If two triangles in

  • SSS: three side lengths or
  • Hours per week: two side lengths and the measure of the included angle or
  • SW: two side lengths, and the degree of the angle opposite the longer side or
  • WSW: one side length and the dimensions of the two adjacent angles or
  • WWS: a side, the measure of an applied and the angle of the opposite

Match, they agree in the other side lengths or angle measurements and are therefore congruent.

Congruence in the spatial geometry

In the solid geometry (spatial geometry) is referred to in polyhedra optionally also of the congruence of the corners, if two corners join the same number of edges and faces having the same angle ( in the same order ); not only the angle must be equal in the side faces of the polyhedron, but also all the angles between the respective pairs of edges. The corner must be possible to convert if necessary by a congruence in the other.

  • Euclidean geometry
  • Synthetic geometry
Euclidean group Translation (geometry) Rotation (mathematics)
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