However, if we notice the similar figures, we see that the corresponding angles are of equal measure, but the sides are not of equal length. In the congruent figures, we can see that all the corresponding sides and angles are of equal measure. However, similar figures may have the same shape, but their size may not be the same.įor example, observe the following triangles which show the difference between congruent and similar figures. Congruent figures have the same corresponding side lengths and the corresponding angles are of equal measure. Hence, the property congruence is one sort of equivalence relation between shapes of a plane.There is a difference between congruent and similar figures. The famous three properties, reflexivity, symmetry and transitivity, together make the notion of equivalence.
This property is called transitivity.įor example, if we apply first a shift, and then a rotation, then the resulting new shape is still congruent to the original 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 symmetry.įor 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 is congruent to another shape, then this other shape is also congruent to the original one.Or, similarly, if the rotation above is not a proper rotation, but only a rotation of angle zero. This property is called reflexivity.įor example, if the shift above is not a proper shift, but only a shift making a motion of length zero. If we leave the original shape alone at its original place, then it is congruent to itself.The relationship, that a shape is congruent to another shape, has three famous properties: More specifically, if a shape is congruent to the original one, then it can be reached by the three activities described above. If we combine the three activities one after the other, then we still get congruent shapes.Even if we take a mirror image of the original shape, then we still get a congruent shape.If we rotate instead of shifting, then we also get a shape 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.The following are a few rules to make new shapes congruent to the original one: Two sides and the angle between them makes two triangles congruent (SAS congruence).All three sides of both triangles are the same (SSS congruence).Two angles and a side not between them are the same on both triangles (AAS congruence).Two angles and the side between them are the same on two triangles (ASA congruence).All equilateral triangles that have the same length of their sides are congruent.All squares that have the same length of their sides are congruent.If one of the object has to change its size, then the two objects are not congruent: they are just called similar.Īs an example, 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 here is permitted).Ĭongruent polygons are polygons that if you fold a regular polygon in half that is a congruent polygon. if one can be moved or rotated so that it fits exactly where the other one is, then the two figures are congruent. This means that two geometrical figures are congruent if one object can be repositioned, rotated or reflected-but not resized-so that it coincides exactly with the other object. More formally, two sets of points are called congruent, if and only if one can be transformed into the other by isometry. In geometry, two figures or objects F and F' are congruent (written as F \cong F') if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. The unchanged properties are called invariants. Note that congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distance and angles. The last triangle is neither similar nor congruent to any of the others. The two triangles on the left are congruent, while the third is similar to them.
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