We explain what congruence is and its differences with coherence. In addition, its sense in geometry and in law.
What is congruence?
We speak of congruence to refer to something that bears a certain logical relationship with its environment or with another specific reference, in a similar way to that expressed by the nouns convenience and coherence, of which it is often used as a synonym.
The word congruence comes from Latin congruent, word formed by voices with (“Next to”) and gruere (“Coincide”), although this etymology is somewhat uncertain, since the verb gruere only records are preserved that associate it with “shouting like a crane” or “imitating the sound of the crane”, which does not seem to make much sense in this context.
In any case, the exact concept of congruence is usually determined by the context in which it is used. For example, in the Law there is talk of congruence when there is conformity between the court’s ruling and the claims of the parties involved in the litigation.
But the meaning of the word changes in the realm of religion, on the other hand, where it expresses God’s ability to act without contravening the free will of human beings, and so on in other areas of knowledge.
Congruence and coherence
Although they are usually used synonymously, these two terms – coherence and congruence – do not have the exact same meaning in all contexts. Both express a logical relationship between two referents, but they differ in a more or less subtle aspect: coherence implies a logical relation of conformity, while congruence implies a logical relation of convenience.
This means that something coherent is something that pursues the same logic, that is part of the same way of thinking or that is unified, consistent with itself. For example, it is consistent for a politician with a conservative affiliation to vote against the changes proposed from the progressive sectors. It is coherent because its theory (its ideology) and its practice (its political decisions) are conditioned.
Instead, something is congruent when it is in accordance with your wishes, convenience or aspirations.
In the same example, if the politician of conservative affiliation has many aspirations to be elected president, it would be congruent on his part to vote in favor of the changes coming from the progressive sectors, that is, from his rivals, if this translates into better and better clearer opportunities to have the necessary support to rise to power. His aspirations (to be elected) and his actions (to win support in unsuspected sectors) are congruent.
Congruence in geometry
In mathematics, specifically in the branch of geometry, the term congruence is used to designate the relationship between two geometric figures that have the same dimensions and the same shape, regardless of their spatial orientation, rotation or reflection, that is, when there is an isometric relationship between them.
Thus, as far as Euclidean geometry is concerned, congruence refers to the arithmetic and algebraic equivalence of the mathematical expressions of two figures. Whereas in analytic geometry it requires that the Euclidean distance between any pair of points of a figure in a Cartesian coordinate system is equal to those of a second figure.
For example, two angles are congruent when a 180 ° rotation about their vertex makes them exactly coincide with each other.
Congruence and similarity of triangles
Two triangles are congruent when they have an isometric relationship to each other, which is expressed mathematically as follows: 
ABC≅DEF (that is: triangle ABC is congruent with triangle DEF). This can happen in any of the following cases:
- AAL or ALA case. Two triangles are congruent when they have equal two angles and the side between them, since knowing two of the angles of a triangle, we can determine the third.
ALA case
AAL case
- LAL case. Two triangles are congruent if they have the same two determined sides and the angle where they touch.
- LLL case. Two triangles are congruent if they have three equal sides.
- ALL case. Two triangles are congruent if they have two equal sides and the opposite angle to that of those sides is also equal. But we must know if it is a right triangle or if its angles are obtuse, first.
Congruence principle
In procedural law, the principle of consistency is known as a maximum that requires the judge of any litigation to reach conclusions that are congruent, that is, consistent, with the requests of the parties in the claim and with the facts recorded in the same.
This means that a judge must make a decision within the aspirations of the parties in dispute, without involving causes unrelated to the case in question and without exceeding the compensation requested by the plaintiff. This means that the judge must operate within the parameters dictated by the case itself.
However, depending on the legal framework of each country, there are specific matters in which the principle of consistency may present exceptions, such as family matters or when it is necessary to provide special protection to one of the parties.