# 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.

## Formula

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}**

## Criteria

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.