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:

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 similarity symbol ($\sim$) 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.

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.

Do you know what the answer is?

Question 1

Look at the triangles in the diagram.

According to which theorem(s) are the triangles congruent?

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 $∢A=30°$

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: SAS

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.

Those that satisfy any triangle congruence criterion.

How to solve triangle congruence problems?

By identifying that according to the criterion used, the corresponding sides or angles are equal.

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Examples and Exercises with Solutions on Congruent Triangles

Exercise #1

Choose the pair of triangles that are congruent according to S.S.S.

Step-by-Step Solution

In answer A, we are given two triangles with different angles, therefore the sides are also different and they are not congruent according to S.S.S.

In answer B, we are given two right triangles, but their angles are different and so are the sides. Therefore, they are not congruent according to S.S.S.

In answer D, we do not have enough data, therefore it is not possible to determine that they are congruent according to S.S.S.

In answer C, we see that all the sides are equal to each other in both triangles and therefore they are congruent according to S.S.S.

Answer

Exercise #2

Given: ΔABC isosceles

and the line AD cuts the side BC.

Are ΔADC and ΔADB congruent?

And if so, according to which congruence theorem?

Step-by-Step Solution

Since we know that the triangle is isosceles, we can establish that AC=AB and that

AD=AD since it is a common side to the triangles ADC and ADB

Furthermore given that the line AD intersects side BC, we can also establish that BD=DC

Therefore, the triangles are congruent according to the SSS (side, side, side) theorem

Answer

Congruent by L.L.L.

Exercise #3

Look at the triangles in the diagram.

Determine which of the statements is correct.

Step-by-Step Solution

Let's consider that:

AC=EF=4

DF=AB=5

Since 5 is greater than 4 and the angle equal to 34 is opposite the larger side in both triangles, the angle ACB must be equal to the angle DEF

Therefore, the triangles are congruent according to the SAS theorem, as a result of this all angles and sides are congruent, and all answers are correct.

Answer

All of the above.

Exercise #4

Look at the triangles in the diagram.

Which of the following statements is true?

Step-by-Step Solution

This question actually has two steps:

In the first step, you must define if the triangles are congruent or not,

and then identify the correct answer among the options.

Let's look at the triangles: we have two equal sides and one angle,

But this is not a common angle, therefore, it cannot be proven according to the S.A.S theorem

Remember the fourth congruence theorem - S.A.A If the two triangles are equal to each other in terms of the lengths of the two sides and the angle opposite to the side that is the largest, then the triangles are congruent.

But the angle we have is not opposite to the larger side, but to the smaller side,

Therefore, it is not possible to prove that the triangles are congruent and no theorem can be established.

Answer

It is not possible to calculate.

Exercise #5

Look at the triangles in the diagram.

Which of the following statements is true?

Step-by-Step Solution

According to the existing data:

$EF=BA=10$(Side)

$ED=AC=13$(Side)

The angles equal to 53 degrees are both opposite the greater side (which is equal to 13) in both triangles.

(Angle)

Since the sides and angles are equal among congruent triangles, it can be determined that angle DEF is equal to angle BAC

Answer

Angles BAC is equal to angle DEF.

Check your understanding

Question 1

Look at the triangles in the diagram.

According to which theorem(s) are the triangles congruent?