Congruence of triangles

In the field of mathematics, two geometric figures can be congruent when these figures have the same dimensions and the same shape regardless of their position or orientation, in other words, if there is an isometry that relates them: a transformation that can be of translation, rotation and / or reflection. The related parts between congruent figures are known as homologous or corresponding figures .

What is the congruence of triangles?

The triangle congruence is when the triangles have the same shape and the same size and are shown to be consistent when their angles corresponding with the same measure and its sides are congruent with each other.

  • Definition of triangle congruence
  • Formula
  • Criteria
  • Properties of triangle congruence
  • Triangle congruence application
  • Examples

Definition of triangle congruence

The term congruence has several meanings depending on the mathematical point of view. For this reason, within the field of Euclidean geometry , congruence is equivalent to a mathematical equality the same as in arithmetic and algebra. In the area of analytic geometry , congruence is defined as two figures determined by points that lie on a Cartesian coordinate system which are congruent with each other.

In the field of mathematics , two point figures can be congruent when they have equal sides and the same size if there is an isometry that relates them. In other words, two figures can become congruent if they have the same shape and size, regardless of whether their position or orientation is different. The parts that coincide in the congruent figures are called homologous or corresponding.


In similar triangles the following conditions and formulas are fulfilled :

  • The homologous angles are equal:

α = α ‘

β = β ‘

ϒ = ϒ ‘

  • The homologous sides are proportional:

a / a ‘=  b / b’ =  c / c ‘= r

A  r  is called  similarity ratio .

It follows that the ratio of the perimeters of two similar triangles is also the  ratio of similarity  and that the ratio of their areas is the  square of the ratio of similarity :

perimeter / perimeter ‘= (a + b + c) / (a’ + b ‘+ c’) = r

area / area ‘= r 2


The criteria that exist for the congruence of triangles are often called by the name of generic congruence postulates or theorems . The criteria for a congruence of triangles tell us that it is not necessary to verify the congruence of the 6 pairs of elements, that is, the 3 pairs of sides and 3 pairs of angles, since we can verify the congruence of three pairs of elements . These criteria are as follows:

  • Two triangles are congruent if two of their angles are equal and respective as is the side between them. In a triangle, if we know two of its angles, the third angle is uniquely determined.
  • Two triangles are congruent if they have two equal sides and the same angle between them.
  • Two triangles are congruent if they have three equal sides .
  • Two triangles are congruent when they have two sides and the angle on one of them of the same measure . This case is not congruent if we do not give more information about the triangle, such as whether it is a right triangle or whether or not it has obtuse angles.

Properties of triangle congruence

The properties that have the congruence of the triangles are the following:

  • Reflective property : it applies to all angles A and tells us that an angle is congruent to itself.

∢A ≅ ∢A

  • Symmetric property : applies to any of the angles A and B.  The order of congruence does not matter.

If ∢A ≅ ∢B find ∢B ≅ ∢A

  • Transitive property : this also applies to any of the angles . If two angles are congruent to a third angle, this means that the first two angles are congruent as well.

If ∢A ≅ ∢B and ∢B ≅ ∢C find ∢A ≅ ∢C

Triangle congruence application

One way we can apply the concept of triangle congruence in everyday life is to establish distances in everyday life, usually when we use a map or diagram as a model. When using congruent triangles to identify distances, the sides of two corresponding triangles must be equated.

Another important application is the creation of geometric figures through constructions or a drawing made only by means of a ruler and compass and they can also be useful in the task of finding distances through the use of congruent triangles.

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