Congruence

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An example of congruence. The two triangles on the left are congruent, while the third is similar to them. The last triangle is neither similar nor congruent to any of the others. Note that congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distance and angles. The unchanged properties are called invariants.

In geometry, two figures or objects are congruent if they have the same shape and size. Also if one has the same shape and size as the mirror image of the other.[1]

More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by isometry. For isometry, rigid motions are used.

This means that one object can be repositioned and reflected (but not resized) so it coincides exactly with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.

Congruent polygons are polygons that if you fold a regular polygon in half that is a congruent polygon.

Two geometrical shapes are congruent if one can be moved or rotated so that it fits exactly where the other one is. If one of the object has to change its size, the two objects are not congruent: they are just called similar.

If two figures or objects are congruent, they have the same shape and size; but they can be rotated, moved, mirror imaged (reflected) or translated, so that it fits exactly were the other one is.

Examples[change | change source]

  • all squares that have the same length of their sides are congruent.
  • all equilateral triangles that have the same length of their sides are congruent.

Tests for congruency[change | change source]

  • Two angles and the side between them are the same on two triangles (ASA congruence)
  • Two angles and a side not between them are the same on both triangles (AAS congruence)
  • All three sides of both triangles are the same (SSS congruence)
  • two sides and the angle between them makes 2 triangles congruent (SAS congruence)

How can we get new congruent shapes?[change | change source]

We have quite a few possibilities, a few rules to make new shapes congruent to the original one.

  • If we shift a geomentrical shape in the plane, then we get a shape which is congruent to the original one.
  • If we rotate instead of shifting, then we also get a shape congruent to the original one.
  • Even if we take a mirror image of the original shape, then we still get a congruent shape.
  • If we combine the three activities one after the other, then we still get congruent shapes.
  • There are no more congruent shapes. More accurately, this means that if a shape is congruent to the original one, then it can be reached by the three activities described above.

The relationship, that a shape is congruent to another shape has three famous properties.

  • If we leave the original shape alone at its original place, then it is congruent to itself. This behaviour, this property is called reflexivity.
For example, if the shift above is not a proper shift, but only a shift making a motion of length zero. Or, similarly, if the rotation above is not a proper rotation, but only a rotation of angle zero.
  • If a shape is congruent to another shape, then this other shape is also congruent to the original one. This behaviour, this property is called symmetry.
For example, if we shift back, or rotate back, or mirror back the new shape to the original one, then the original shape is congruent to the new one.
  • If a shape C is congruent to a shape B, and the shape B is congruent to the original shape A, then the shape C is also congruent to the original shape A. This behaviour, this property is called transitivity.
For example, if we apply first a shift, and then a rotation, then the resulting new shape is still congruent to the original one.

The famous three properties, reflexivity, symmetry and transitivity together make the notion of equivalence. Hence, the property congruence is one sort of equivalence relation between shapes of a plane.

References[change | change source]

  1. Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Congruent Figures" (PDF). Addison-Wesley. p. 167. Retrieved September 2013. Check date values in: |accessdate= (help)