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≅ΔDEF 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(=) since the sides are respectively equal.
And above it, a tilde (∼) 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, ∼
Symbol for twoparallel lines, ∥
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≅ΔDEF 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≅ΔDEF
We also know that theangles
∢E=60°
∢A=51°
Additionally, we have the following data about the sides:
AB=5cm
AC=4cm
EF=3.9cm
Find the angles∢B,∢C,∢D,and∢F And then find the length of the sidesBC,DE andDF
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=5cm
AC=DF=4cm
EF=BC=3.9cm
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≅ΔEFD
ΔABC≅ΔFDE
ΔABC≅ΔDEF
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≅ΔDEF
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 thatFE=4cm
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≅ΔABCBy 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 form2 congruent triangles, and the sideAC=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
AE=BE=5cm
Show thatΔBED≅ΔAECand thatAC=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
Test your knowledge
Question 1
Point K is located in the middle of AC.
\( ∢A=∢C \)
According to which theorem are the triangles Δ AMK and Δ CBK congruent?
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)
△DEF≅△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∥DC (they are parallel)
AB=DC
Prove thatAO=CO, and also thatBO=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≅ΔABC According to the congruence criterion:side, side, side (SSS)
Example
In a quadrilateralABCD:
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.