Congruent Triangles

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.

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.

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

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 the angles

$∢E=60°$

$∢A=51°$

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:

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

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

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

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.

If we know that in a triangle $\triangle ABC$

$AB=AC=BC$

And in a triangle $\triangle EFD$

$DF=DE=EF$

And we also know that $AB=EF$

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:

$ΔABC\congΔEFD$

$ΔABC\congΔFDE$

$ΔABC\congΔDEF$

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$FE=4\operatorname{cm}$

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.

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.

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 $AC=BD$

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.

In the following drawing, we know that:

• $AB\parallel DC$ (they are parallel)
• $AB=DC$

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

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)

In a quadrilateral $ABCD$:

• $AB=AD$
• $CB=CD$

Prove that the angles $∢D=∢B$

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:

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

Therefore:

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

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

• $AE=CD(L)$
• $∢AEC=∢CDA=90°$

Prove that $AB=BC$

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.

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