Congruent triangles are identical triangles.

That means in triangles whose angles and sides are equal, their area and perimeter 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:

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

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Test yourself on congruent triangles!

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Look at the triangles in the diagram.

Which of the statements is true?

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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 have two triangles Δ ABC Δ DEF

We know that the sides

  • AB=DE AB=DE
  • AC=DF AC=DF
  • BC=EF BC=EF

We also know that the following angles are equal:

  • A=D ∢A=∢D
  • B=E ∢B=∢E
  • C=F ∢C=∢F

Therefore, we can deduce the following:

ΔABCΔDEF 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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Basic Terms in Triangle Congruence

  • 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 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 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 two parallel lines, \parallel
  • In every triangle, the sum of its interior angles is equal to 180° 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.

Do you know what the answer is?

For example

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

DEF and the congruence is noted according to the order of the vertices

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

A=D ∢A=∢D

B=E ∢B=∢E

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

AC=DF AC=DF

AB=DE 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 ΔABC and ΔDEF ΔDEF , are congruent, in the order of the vertices, that is ΔABCΔDEF ΔABC\congΔDEF

We also know that the angles

E=60° ∢E=60°

A=51° ∢A=51°

Additionally, we have the following data about the sides:

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

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

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

2 triangles with the following data about the sides

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

Since the triangles are congruent, we know that:

E=B=60° ∢E=∢B=60°

A=D=51° ∢A=∢D=51°

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

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

So:

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

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

EF=BC=3.9cm 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° 60° . As we have already mentioned, in every triangle there are three angles whose sum gives us 180° 180° . Therefore, 2 2 equilateral triangles that have a side of equal length will be congruent to each other.


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For example

If we know that in a triangle ABC \triangle ABC

AB=AC=BC AB=AC=BC

And in a triangle EFD \triangle EFD

DF=DE=EF DF=DE=EF

And we also know that AB=EF AB=EF

if we know that in a triangle ABC

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

ΔABCΔEFD ΔABC\congΔEFD

ΔABCΔFDE ΔABC\congΔFDE

ΔABCΔDEF 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:

  • ΔABCΔDEF ΔABC\congΔDEF
  • AB=AC AB=AC
  • DE=DF DE=DF
  • D=30° D=30°
example with isosceles triangle

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

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

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


Do you think you will be able to solve it?

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° 180° , since as we've mentioned, in every triangle the sum of the interior angles is always equal to 180° 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 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 AB=DEL
  • B=EA ∢B=∢EA
  • CB=FEL CB=FEL
Congruence postulates new

Therefore:

ΔDEFΔABC ΔDEF\congΔABC By the congruence criterion: SAS SAS

From which we can deduce:

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

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


For example

Show that when two lines intersect, they form 22 congruent triangles, and the side AC=BD 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=4cm DE=CE=4\operatorname{cm}

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

example with the following information

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

AssertionArgument
  • 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


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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 ∢D=∢A (angle)
  • DE=AB DE=AB (side)
  • E=B ∢E=∢B (angle)
ABC congruence postulate angle, side, angle (ASA)

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

From which we can deduce:

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

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

Example

In the following drawing, we know that:

  • ABDC AB\parallel DC (they are parallel)
  • AB=DC AB=DC
Prove that AO=CO

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


AssertionArgument
  • 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 DE=AB (side)
  • DF=AC DF=AC (side)
  • EF=BC EF=BC (side)
triangles with all three sides respectively equal

Therefore:

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

Example

In a quadrilateral ABCD ABCD :

  • AB=AD AB=AD
  • CB=CD CB=CD
a quadrilateral ABCD

Prove that the angles D=B ∢D=∢B


AssertionArgument
  • 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

Do you know what the answer is?

Fourth Criterion for Congruence - Side, Side, Angle (SSA)

Definition: Two triangles that have 2 equal sides, as well as the angle opposite to the larger of those sides, will be congruent.

From this, the following criterion is derived: Two right triangles that have the hypotenuse and one leg of the same length will be congruent.

Data:

  • BA=ED BA=ED (side)
  • BC=EF BC=EF (side)
  • BAC=EDF ∢BAC=∢EDF (angle) 
  • the following relationship is given BC>AB BC>AB (side)
right triangles

Therefore:

DEFABC \triangle DEF\cong\triangle ABC This is due to the criterion: side, side, angle (S.A.S)

Example

In triangle ABC \triangle ABC , two of its heights are equal.

  • AE=CD(L) AE=CD(L)
  • AEC=CDA=90° ∢AEC=∢CDA=90°
In triangle ABC, two of its heights are equal

Prove that AB=BC AB=BC


AssertionArgument
  • AC=AC (Side)
  • AE=CD (Side)
  • AEC=CDA=90o

therefore

  • Δ CAE Δ ACD
  • C=A
  • AB=BC
  • Common side
  • Data
  • Data - The largest angle in the triangle

Therefore
according to the side, side, angle congruence postulate: We verify

We verify
A triangle whose base angles are equal is an isosceles triangle

Questions about the topic

How to prove that a triangle is congruent?

By using one of the triangle congruence criteria, for example, SSA (side, angle, side).


Which pairs of triangles are congruent?

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.



Check your understanding

A brief visual summary of the article:


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Examples and Exercises with Solutions on Congruent Triangles

Exercise #1

Look at the triangles in the diagram.

Which of the following statements is true?

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Step-by-Step Solution

According to the existing data:

EF=BA=10 EF=BA=10 (Side)

ED=AC=13 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.

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?

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Step-by-Step Solution

As we know the triangle is isosceles, then AC=AB

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

Given that the line AD intersects side BC, and therefore BD=DC

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

Answer

Congruent by L.L.L.

Exercise #3

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

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Exercise #4

Look at the triangles in the diagram.

Determine which of the statements is correct.

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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, then the angle ACB is 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 #5

The triangles ABO and CBO are congruent.

Which side is equal to BC?

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Video Solution

Step-by-Step Solution

Let's consider the corresponding congruent triangles letters:

CBO=ABO CBO=ABO

That is, from this we can determine:

CB=AB CB=AB

BO=BO BO=BO

CO=AO CO=AO

Answer

Side AB

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