That means intriangles whoseangles and sides are equal, theirarea andperimeter will also be equal.

But keep in mind that this case is different from when triangles are similar, that is, when the angles are equal but the side lengths are different in the corresponding ratio.

Congruence Criteria

To prove that 2 triangles are congruent we can use one of the following criteria:

SAS - side, angle, side

ASA - angle, side, angle

SSS- side, side, side

SSA- side, side, angle

By verifying one of the triangle congruence criteria, we can affirm that the triangles are congruent.

When we talk about triangles, we can find different types of matches. There are triangles that are equal only in their angles and are called similar triangles, and there are triangles that are equal in both their angles and sides, being identical to each other. We will call the latter congruent triangles, and we will learn about them in this article.

Congruent Triangles

First, let's start with an example of congruent triangles:

We know that the sides

$AB=DE$

$AC=DF$

$BC=EF$

We also know that the following angles are equal:

$∢A=∢D$

$∢B=∢E$

$∢C=∢F$

Therefore, we can deduce the following:

$ΔABC\congΔDEF$ According to the order of the vertices

Look at the following symbol: $≅$ In mathematics, it means congruence, and if you look closely, you'll see that it is composed of two symbols

the equal sign$\left(=\right)$ since the sides are respectively equal.

And above it, a tilde ($\sim$) which itself represents similarity both in mathematics, and among different triangles whose angles will be equal.

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Triangle - a geometric figure determined by three line segments (three sides), or by three non-collinear points called vertices.

Vertices of a triangle, These are the points of intersection between line segments. The vertices are represented with capital letters. For example $A,B,C$.

Side of the triangle is the line segment that connects 2 vertices of a triangle, and we denote them as $AB,CB$, etc.

Similar triangles, These are triangles whose corresponding angles are equal, but their sides have different lengths.

Symbol of similarity, $\sim$

Symbol for twoparallel lines, $\parallel$

In every triangle, the sum of its interior angles is equal to $180°$

Congruent Triangles

Writing a congruence should be done according to the order of the vertices representing the angles that are equal to each other, so that the first letter of the first triangle will correspond to the first letter of the second triangle where both angles are equal. The second letter of the first triangle will correspond to the second letter of the second triangle where both angles are equal, and finally, the third letter in both triangles will indicate that the angles are equal.

It's important to remember that when we have congruent triangles, we will always have equal sides opposite the equal angles.

Given that $ΔABC\congΔDEF$ and the congruence is noted according to the order of the vertices.

Therefore, we can deduce: That the equal angles are:

$∢A=∢D$

$∢B=∢E$

$∢C=∢F$ Therefore, the equal sides are: $BC=EF$

$AC=DF$

$AB=DE$

Before continuing, and to confirm that we have understood, let's look at the following example of a question about congruent triangles, and try to solve it.

Given that the triangles $ΔABC$ and $ΔDEF$, are congruent, in the order of the vertices, that is $ΔABC\congΔDEF$

We also know that theangles

$∢E=60°$

$∢A=51°$

Additionally, we have the following data about the sides:

$AB=5\operatorname{cm}$

$AC=4\operatorname{cm}$

$EF=3.9\operatorname{cm}$

Find the angles$∢B,∢C,∢D, and ∢F$ And then find the length of the sides$BC$,$DE$ and$DF$

Since the triangles are congruent, we know that:

$∢E=∢B=60°$

$∢A=∢D=51°$

Therefore, the answer regarding the remaining angles is $∢F=∢C=69°$.^{ }Since the total sum of the angles of a triangle is $180°$.

The same will also apply to the sides, as these are congruent triangles.

So:

$AB=DE=5\operatorname{cm}$

$AC=DF=4\operatorname{cm}$

$EF=BC=3.9\operatorname{cm}$

Congruence of Equilateral Triangles

For triangles whose sides are equal, their angles will also be equal. Since opposite equal sides, we have equal angles, and therefore each angle of an equilateral triangle measures $60°$. As we have already mentioned, in every triangle there are three angles whose sum gives us $180°$. Therefore,$2$ equilateral triangles that have a side of equal length will be congruent to each other.

We can conclude that all sides are equal, and that each angle in these triangles measures $60°$

Therefore, we can establish that two equilateral triangles that have one side of equal length and regardless of the order in which the vertices are listed, will be congruent to each other.

For example:

$ΔABC\congΔEFD$

$ΔABC\congΔFDE$

$ΔABC\congΔDEF$

Triangle Congruence in Isosceles Triangles

An isosceles triangle has two sides of equal length and the two angles opposite to the equal sides are also equal in measure.

In the following example, we have these pieces of information:

$ΔABC\congΔDEF$

$AB=AC$

$DE=DF$

$D=30°$

From this information, we can conclude that angle $∢A=30°$

And therefore the angles $∢C=∢B=∢F=∢E=75°$

We can also conclude that$FE=4\operatorname{cm}$

Do you think you will be able to solve it?

Question 1

Given the triangles in the drawing

According to which theorem, are the triangles congruent?

What is the minimum amount of data needed to verify if there is triangle congruence?

Initially, five pieces of information might be enough to prove that two triangles are congruent:

3 equal sides

2 equal angles (because the additional angle will always complete $180°$, since as we've mentioned, in every triangle the sum of the interior angles is always equal to $180°$).

But sometimes we can know that triangles are congruent with just three pieces of information. For this, it's necessary to know the congruence criteria, which describe different possibilities for the congruence of triangles with only $3$ pieces of information.

Triangle Congruence Criteria

First Criterion: Side, Angle, Side.

Which we will abbreviate with the following initials: LAL

Definition: Two triangles are congruent if two of their sides have the same length as two sides of the other triangle, and the angles included between those sides are also equal.

Given:

$AB=DEL$

$∢B=∢EA$

$CB=FEL$

Therefore:

$ΔDEF\congΔABC$By the congruence criterion:$SAS$

From which we can deduce:

$BC=FE$ are equal sides in congruent triangles, as well as the sides $AC=DF$ are equal for the same reason.

It can also be concluded that the angles $∢C=∢F$ are equal angles in congruent triangles.

For example

Show that when two lines intersect, they form$2$ congruent triangles, and the side$AC=BD$

To do this, we must set out the information in the following order:

The information we have

What we want to prove

This way, you can develop the reasoning process, and the explanation of what you want to demonstrate.

Here are the following pieces of information:

$DE=CE=4\operatorname{cm}$

$AE=BE=5\operatorname{cm}$

Show that$ΔBED\congΔAEC$and that$AC=BD$

Assertion

Argument

BE=AE=5 (Side)

∢DEB=∢AEB (Angle)

DE=CE=4 (Side)

Therefore

Δ BED ≅Δ AEC

AC=BD

Data

Vertically opposite angles

Data

Therefore, according to the side-angle-side congruence postulate

We verify Corresponding sides in superimposed triangles are equal

Second Criterion of Congruence - Angle, Side, Angle (ASA)

Definition:

Two triangles are congruent if two angles and the included side are equal (in both length and degrees) to two angles and the included side of the other triangle.

We have the following data:

Therefore:

$∢D=∢A$ (angle)

$DE=AB$ (side)

$∢E=∢B$ (angle)

$\triangle DEF\cong\triangle ABC$ According to the congruence criterion:angle, side, angle (ASA)

From which we can deduce:

$BC=FE$ are equal sides in congruent triangles, and so are the sides $AC=DF$ (for exactly the same reason).

And we can also draw the following conclusion $∢C=∢F$ are equal angles in congruent triangles.

Example

In the following drawing, we know that:

$AB\parallel DC$ (they are parallel)

$AB=DC$

Prove that$AO=CO$, and also that$BO=DO$

Assertion

Argument

DC || AB

Therefore

∢C=∢A (Angle)

∢D=∢E (Angle)

DC=AB (Side)

Therefore

Δ CDO ≅ Δ ABO

CO=AO and also DO=BO

Data

Therefore

Alternate interior angles between parallel lines

Alternate interior angles between parallel lines

Data

Therefore According to the angle-side-angle congruence postulate

We verify Common sides in overlapping triangles

Third Criterion of Congruence - Side, Side, Side (SSS)

Definition: In these triangles, all three sides are respectively equal.

Data:

$DE=AB$ (side)

$DF=AC$ (side)

$EF=BC$ (side)

Therefore:

$ΔDEF\congΔABC$ According to the congruence criterion:side, side, side (SSS)

Example

In a quadrilateral$ABCD$:

$AB=AD$

$CB=CD$

Prove that the angles$∢D=∢B$

Assertion

Argument

AB=AD (Side)

BD=CD (Side)

AC=AC (Side)

Therefore

Δ ADC ≅Δ ABC

∢D=∢B

Data

Data

Common side

Therefore, according to the congruence postulate: side, side, side

We verify Corresponding angles in overlapping triangles are equal

Examples and Exercises with Solutions on Congruent Triangles

examples.example_title

Given the congruent triangles ABC and CDA

Which angle is equal to angle BAC?

examples.explanation_title

We observe the order of the letters in the congruent triangles and write the matches (from left to right).

$ABC=CDA$

That is:

Angle A is equal to angle C

Angle B is equal to angle D

Angle C is equal to angle A

From this, it is deduced that angle BAC (where the letter A is in the middle)

Is equal to angle C, that is, to angle DCA (where the letter C is in the middle)

examples.solution_title

C

examples.example_title

Are the triangles shown in the drawing congruent? If so, explain according to which congruence theorems

examples.explanation_title

To answer the question, we need to know the fourth congruence theorem: S.A.S.

The theorem states that triangles are congruent when they have a pair of sides and an angle equal

However, there is a condition: the angle must be opposite the larger side of the triangle.

We start with the sides:

DF=CB=16 GD=AC=9

Now, we look at the angles:

A=G=120

We know that an angle of 120 is an obtuse angle and this type of angle is always opposite the larger side of the triangle.

Therefore, we can argue that the triangles are congruent according to the S.A.S theorem

examples.solution_title

Congruent according to S.A.S

examples.example_title

Are the triangles in the image congruent?

examples.explanation_title

Although the lengths of the sides are equal in both triangles, we observe that in the right triangle the angle is adjacent to the side whose length is 7 and in the triangle on the left side the angle is adjacent to the side whose length is 5.

Since it's not the same angle, the angles between the triangles do not match and therefore the triangles are not congruent.

examples.solution_title

No

examples.example_title

Which of the triangles are congruent?

examples.explanation_title

Let's observe the angle in each of the triangles and note that each time it is opposite to the length of a different side.

Therefore, none of the triangles are congruent since it is impossible to know from the data.

examples.solution_title

It is not possible to know based on the data

examples.example_title

What data must be added for the triangles to be congruent?

examples.explanation_title

Let's consider that:

DF=AC=8

DE=AB=5

8 is greater than 5, therefore the angle DEF is opposite the larger side and is equal to 65 degrees.

That is, the figure we are missing is the angle of the second triangle.

We will examine which angle is opposite the large side AC.

ABC is the angle opposite the larger side AC so it must be equal to 65 degrees.