Similar triangles are triangles whose three angles are equal respectively and also the ratio between each pair of corresponding sides is equal. Two similar triangles are actually larger or smaller versions each other.

The ratio of similarity is the ratio between two corresponding sides in two similar triangles.

To prove similarities betweentriangles, we will use the following theorems:

Angle-Angle (A.A): If two angles are equal respectively between two triangles, then the triangles are similar.

Side-Angle-Side (S.A.S): If the ratio of two pairs of sides is equal, and also the angles between them are equal to each other, then the triangles are similar.

Side-Side-Side (S.S.S.): If for two triangles, the ratio of the three sides in one triangle to the three pairs in the other triangle is equal (similarity ratio), then the triangles are similar.

For similarity of polygons we will define it this way: if for two polygons all angles are equal and there is a constant ratio between two corresponding sides, then the polygons are similar.

Intuitively, just like similar triangles, also two similar polygons are actually an enlargement or reduction of each other.

Definition: Similar triangles are triangles whose three angles are equal respectively and also the ratio of each pair of corresponding sides is equal. Two similar triangles are actually an enlargement or reduction of each other.

To understand this, let's look at the following example:

Example 1

Given the two triangles in the drawing

$Δ ABC$ $Δ DEF$

Given that the triangle $Δ ABC$ and the triangle $Δ DEF$ are similar triangles. We will mark this with the sign $~$

It looks like this: $Δ ABC$ ~ It is important to write the correct order of the vertices, similar to the superposition of triangles. $Δ DEF$

From here we can conclude that the three angles are equal respectively, ie:

$∢A=∢D$ $∢B=∢E$ $∢C=∢F$

And we can conclude that the ratio between each pair of corresponding sides is equal. That is:

$\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}$

This ratio of sides is called the ratio of similarities. It is important to note that two overlapping triangles are also similar triangles when the ratio of similarity is $1$.

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The similarity ratio is the ratio between two corresponding sides in two similar triangles.

Example 2

Since the two triangles $Δ ABC$ and $Δ DEF$ are similar triangles, that is:

$Δ ABC$ ~ $Δ DEF$

It is also given $AB = 8$ $BC = 12$ $CA = 6$ In addition: $DE = 4$ $EF = 6$ $FD = 3$ All the data are marked in the drawing.

Calculate the similarity ratio between the two triangles.

Pay attention that we do not know the size of the angles, but we do not need it to calculate the similarity ratio, since it is stated that they are similar triangles, so their respective angles measure the same. We can calculate the similarity ratio by the ratio between each pair of corresponding sides:

That is, we have seen that the similarity ratio between the triangle $Δ ABC$ For the triangle $Δ DEF$ Is $1:2$. QED

Pay attention to the similarity ratio between the lengths of the sides of the triangle $Δ DEF$ For the triangle $Δ ABC$ It is $2:1$ Intuitively, the length of each side in a triangle $Δ ABC$ is $2$ times longer than each side in a triangle $Δ DEF$ Respectively.

To prove the similarity between triangles we will use one of the following three theorems: - Angle-Angle (A.A): If two angles are equal in correspondence between two triangles, then the triangles are similar. - Side-Angle-Side (S.A.S): If the ratio between two pairs of sides is equal, and also the angles included between them are equal to each other, then the triangles are similar. - Side-side-side (S.S.S.): If for two triangles, the ratio between the three sides in one triangle to the three pairs in the other triangle is equal (similarity ratio) then the triangles are similar.

Exercise with example - (How to calculate the length of the side)

Given two triangles in the drawing below $Δ ABC$ $Δ DEF$ are similar triangles, i.e. $Δ ABC$ ~ $Δ DEF$ In addition given: $AB = 5$ $DE = 2.5$ $FD = 1$ $∢A=∢D$ $∢B=∢E$ $∢C=∢F$

All data are labeled in the drawing.

Question: what is the length of the side $AC$?

Solution: The two triangles are similar, so we will calculate the similarity ratio and use it to solve the assignment. Remember that the ratio between two sides in similar triangles is equal and therefore:

$\frac{AB}{DE}=\frac{5}{2.5}=\frac{2}{1}$

That is, the similarity ratio is $2:1$, and each side of the triangle $\triangle ABC$ is twice as large as any corresponding side in the triangle $\triangle DEF$. Now we can calculate the length of side AC. According to the similarity ratio:

$\frac{AC}{DF}=2$

We replace and obtain:

$\frac{AC}{1}=2$

That is, we obtained:

$AC=2$

QED

Do you know what the answer is?

Question 1

What is the ratio between the sides of the triangles ΔABC and ΔMNA?

Definition: If for two polygons all angles are equal and there is a constant ratio between two corresponding sides, then the polygons are similar. Intuitively, as in similar triangles, two similar polygons are actually larger and smaller versions of each other.

Example 3 - Similar polygons

These two squares are similar squares:

Any two corresponding angles are equal since all angles are equal. The ratio of the two corresponding sides, i.e., the similarity ratio, is $2/1$ or, in other words, each of the sides is twice as large for the large square as for the small square.

Check your understanding

Question 1

The ratio of the areas of similar triangles is \( \frac{9}{100} \)Given that the perimeter of the large triangle is 129 cm, what is the perimeter of the small triangle?

Two triangles are similar if their three respective angles have the same measure and the ratio between the pairs of their respective sides is the same.

What are the three criteria for similarity of triangles?

There are three criteria to determine if two triangles are similar or not, which are the following:

Side-Side-Side (SSS): If the ratio of their three pairs of corresponding sides is the same then two triangles are similar.

Side-Angle-Side (SAS): Two triangles are similar if the ratio of two pairs of corresponding sides is the same and the angle between these two pairs is the same, then they are similar triangles.

Angle-Angle (AA): For two triangles to be similar by this criterion, two of their respective angles must measure the same and therefore the third angle must also have the same measure as the angle corresponding to that angle. That is, their three corresponding angles measure the same.

How to find the similarity ratio?

In order to get the ratio of similarity we must calculate the ratio or division of each of their respective pairs of sides, and this relationship is the same for each of the pairs of sides when two triangles are similar.

When are two triangles similar by the SSS criterion?

Two triangles are similar when they have the same shape but their sides do not necessarily have to measure the same, they are similar by the SSS criterion as long as their corresponding sides are proportional, i.e. the ratio between their sides is the same.

What is the similarity of polygons?

In the same way that the similarity of triangles, two polygons are similar when they have the same shape, they do not necessarily have to have the same measure in their sides, that is, their corresponding angles are equal and the ratio of their corresponding sides is the same for all of them.

Check your understanding

Question 1

What is the ratio between the sides of the triangles ΔABC and ΔMNA?