In right triangles, we have a condition that already exists in the first place. It refers to the right angle that iss given and that turns a triangle into a right triangle.

In the second stage, we will move on to the sides. In every right triangle we have two perpendiculars (two sides between which the right angle is comprised) and the other (the larger side of the triangle that faces the right angle).

When there are two right triangles in front of us, in which one size is perpendicular and the size of the rest is equal to each other, then we can conclude that these are congruent triangles.

Right triangle congruence takes into account the unique properties of right triangles and uses them to prove congruence.

We are already familiar with the usual congruence theorems:

Congruence according to Side-Angle-Side Congruence according to Angle-Side-Angle Congruence according to Side-Side-Side.

We will illustrate this with an example.

The graph shows two right triangles: $\triangle ABC$ and $\triangle DEF$.

Both triangles have a right angle (equal to $90^o$ degrees).

Moreover, in both triangles there is a perpendicular equal to $3$ (i.e., $AB = DE$), while the remaining one is equal to $5 (AC = DF)$.

If we were now to use the Pythagorean theorem, we would reach the size of the second perpendicular in each of the triangles and this perpendicular would come out equal to $4$, since it is the same calculation.

Therefore, we can always make use of the conclusion we have already reached, according to which when we are given two right triangles, in which one of them is perpendicular and the rest are equal to each other, respectively, we can conclude that these are congruent triangles.

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A right triangle is a figure that has three sides and has a right angle, that is, an angle of $90\degree$, like the one shown in the figure.

The $\triangle ABC$ is a right triangle.

What is right triangle congruence?

Recall that the congruence of figures refers when two figures have the same shape and their corresponding sides and angles are equal, in the case of right triangles, it must be exactly the same. The difference here is that right triangles already have a defining characteristic that identifies them. If we have two right triangles then we already know that one of its angles measures $90\degree$ and it is only a matter of seeing what congruence criteria is met to verify that they are congruent triangles.

What are the congruence criteria to determine if two right triangles are congruent?

There are four criteria to determine if two triangles are congruent or not, which are the following:

SAS- Side, Angle, Side: Two triangles are congruent when two of their sides and the angle between them measure the same.

ASA- Angle, Side, Angle: Two triangles are congruent when two of their corresponding angles and the side between them measure the same.

SSS- Side, Side, Side: Two triangles are congruent if their three corresponding sides measure the same.

SSA- Side, Side, Angle: Two triangles are congruent if two of their corresponding sides and an angle opposite one of them have the same measure.

When are two right triangles not congruent?

Two right triangles are not congruent when they do not meet any of the above congruence criteria, i.e., their corresponding sides and angles have different measures (they have different shapes).