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Congruence is an important concept in geometry: learn how to decide if two triangles are congruent with our triangle congruence calculator.

Here you will learn:

  • What congruence is;
  • Why congruence in triangles is particular;
  • The four main types of congruence in triangles; and
  • How to use our triangle congruence calculator.

What is congruence?

Congruence is a property of geometric shapes (polygons). Take two polygons: we say that they are congruent if all the corresponding sides and angles are congruent.

Why do we need both sides and angles?

  • Sides fix the perimeter of a polygon, but not necessarily the angles: think of deforming a square. We say that they fix the scale of the polygon.
  • Angles fix the shape but not the scale of a polygon. You don't have information about the size of the shape.
  • A combination of both fixes both the shape and the size. We can define congruence only if angles and sizes are defined.

And what is a triangle?

Triangles are the simplest polygons, with only three sides and three angles. Behind this apparent lack of complexity, they hide many interesting properties and relations, many of them reflected in the way we can analyze their congruence.

We identify the sides of a triangle with the letters aa, bb, and cc, while we mark the angles with the corresponding letters of the Greek alphabet: α\alpha, β\beta, and γ\gamma.

Congruence in triangles: calculate identical triangles.

Triangles are the only shapes that don't require a full set of data to assess their congruence. Some strategies allow you to do so with as little as three quantities. The possible ways to determine if two triangles are congruent are:

  • SSS, when three sides are given;
  • SAS, when you know two sides and the angle between them;
  • ASA, when you know two angles and the side between them; and
  • AAS, when you know two angles and one of the adjacent sides.

We will now define the congruence in triangles using these four possible criteria.

SSS triangles congruence

You can determine if two triangles are congruent by knowing just the sides. This interesting property stems from the fact that triangles are non-deformable. Apart from making them the most common shape in architecture and engineering, this property allows us to say that if the sides of two triangles are congruent in pairs, then the two triangles are congruent.

J. Hadamard gave the best proof of this theorem — in our opinion.

SAS triangles congruence

If two triangles have a pair of congruent sides, and the angle between them has the same amplitude, then they are congruent.

🙋 Euclid proved the SAS triangle congruence theorem using the superposition method, which only involves a "mental" exercise in placing one shape over the other. You can see that this is not a strong mathematical argument, and in fact, the proof is not accepted today. The SAS congruence theorem is often considered a postulate.

ASA triangles congruence

An ASA triangle is defined by a set of two angles (say α\alpha and β\beta) and the side between them (in this case, cc). We can prove the triangles' congruence, but only assuming that the SAS theorem is correct.

Take two triangles: ABCA \overset{\triangle}{B} C and DEFD \overset{\triangle}{E} F. We know the angles AB^C=DE^FA\hat{B}C=D\hat{E}F and CA^B=FD^EC\hat{A}{B}=F\hat{D}E, and that the side between them is the same in the two triangles: ABˉ=DEˉ\bar{AB}=\bar{DE}.

Assume that the sides ACˉ\bar{AC} and DFˉ\bar{DF} are different (ACˉ>DFˉ\bar{AC}>\bar{DF}). We can identify a point GG on ACˉ\bar{AC} where the SAS theorem hold, since AGˉ=DFˉ\bar{AG}=\bar{DF}, AB^C=FD^EA\hat{B}C=F\hat{D}E, and ABˉ=DEˉ\bar{AB}=\bar{DE}.

This fixes the other angle (AB^CA\hat{B}C) to the value of DE^FD\hat{E}F. At this point you can see that given an ASA set of data, the triangles must be congruent.

AAS triangles congruence

If you have two triangles with an identical pair of angles and a congruent side not comprised between the angles, the two triangles are congruent. You just defined an AAS triangle.

You can prove the congruence of a triangle of which you know such data by starting with the ASA triangle proof.

When size doesn't matter: similar triangles

Pairs of triangles can be compared using angles instead of sides. In this case, we will miss information on the scale of the polygon: an equilateral triangle as big as a galaxy will look identical to an equilateral triangle as small as a molecule if you look from the right distance. This concept is called by mathematicians similarity.

Triangles with the same set of angles are similar. If the sides also coincides, then we deal with congruent triangles.

In an interesting contrast, knowing the sides is enough to calculate the angles, but the inverse is not true: the angles only fix the shape.

How to use our triangle congruence calculator

Our triangle congruence calculator implements all four possible triangle congruence theorems and crosses them. This means that not only you will be able to compare the same type of input (e.g. SSS and SSS), but also different sets of data (e.g. ASA with SAS): we will perform the needed calculations in the background.

In our calculator, you can see two sets of variables, one for each triangle.

Choose the set of data you know in the variable given for each of them. Proceed to insert the data. If the combination is "legal", we will tell you if the pair of triangles you are studying is congruent or not. If not, we will give you another information: if the triangles you are analyzing are similar.

Other triangle calculators


Are AAA triangles congruent?

AAA triangles are defined by the three interior angles. AAA triangles are not necessarily congruent. Triangles with the same set of angles have the same shape, but not necessarily the same size: the scale of a shape is not determined by the angles. To fix the shape univocally, you need to set at least one side.

Is a triangle with sides a = 1, b = 2, and c = 3 congruent with an ASA triangle with ß = 46.57°, γ = 104.48°, and a = 2?

Yes! To check this congruence, you need to calculate some quantities first. You can decide to:

  1. Calculate the sides of the ASA triangle; or
  2. Calculate the angles of the SSS triangle.

Let's calculate the sides.

  1. Compute the angle α:
    α = 180° - ß - γ= 28.96°
  2. Use the law of sines to compute b and c:
    b = (a/sin(α)) × sin(ß) = 3
    c = (a/sin(α)) × sin(γ) = 4
  3. The two triangles are SSS congruent!

Is SAS enough to calculate the triangle congruence?

Yes. If you know a set of two sides and the angle between them (SAS), then you can determine if a pair of triangles are congruent. This property has been known since the time of Euclid (he gave a rather unsatisfying proof). To this day, mathematicians accept this theorem as an axiom of geometry.

Is SAA the same as AAS?

Yes. The AAS postulate to determine the triangle convergence is the same as SAA: the only requirement is to know two angles and one of the adjacent sides. If you knew the last side, you would have to use the ASA postulate.

Is SSA enough to calculate two triangles congruence?

No. Fixing two sides and one angle gives space to a degree of uncertainty: it is possible to rotate the first side in two positions, one at an acute angle and the other at an obtuse angle. In this case, the scale of the polygon is fixed, but not the shape.

Davide Borchia
Triangle 1
three sides (SSS)
Triangle area: triangle with base and height
Triangle 2
three sides (SSS)
Triangle area: triangle with base and height
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