Similar triangles are triangles for which there is a certain similarity ratio, that is, each of the sides of one triangle is in uniform proportion to the corresponding side in the other triangle. In addition, the angles at the same locations are also equal for the two similar triangles.

How do you prove the similarity of triangles?

To prove the similarity of triangles it is common to use one of three theorems:

Angle-angle (i.e., two pairs of equal angles in triangles).

Side-angle-side (similarity ratio of two pairs of sides in triangles and the angles trapped between them are equal)

Side-side-side-side (similarity ratio of three pairs of sides in triangles).

Similarities of triangles are expressed with the sign $∼$.

The drawing before us shows two similar triangles,$\triangle ABC$ and $\triangle KLM$.

The similarity ratio of the triangles is $2$. This means that each side in the larger triangle $\triangle ABC$ is twice as large as the corresponding side in the smaller triangle $\triangle KLM$.

In addition, the angles at the corresponding places in the two triangles are equal to each other.

As illustrated in the drawing, the following is true:

The angle $\sphericalangle A$ is equal to the angle $\sphericalangle K$ The angle $\sphericalangle B$ is equal to the angle $\sphericalangle L$ The angle $\sphericalangle C$ is equal to the angle $\sphericalangle M$

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We can say that two triangles are similar when they have the same shape even if they have different sizes, for that they must meet some of the similarity criteria.

What are the three similarity criteria?

To know that two triangles are similar they must meet some of the three similarity criteria:

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.

What is the ratio of similarity of two triangles?

It is the ratio between the corresponding sides of those triangles.

How to find the similarity ratio of two triangles?

The similarity ratio is obtained by dividing the corresponding sides of two similar figures, in this case of two triangles.

Let's see an example:

Task

Given the following similar triangles $\triangle ABC\sim\triangle DEF$

Calculate the similarity ratio

Given that $\triangle ABC\sim\triangle DEF$ by the similarity criterion AA.

Then we must locate which are the corresponding sides, and from here we deduce that

$\sphericalangle A=\sphericalangle D$

$\sphericalangle B=\sphericalangle E$

Then the corresponding sides are $AB$ and $DE$

Now to calculate the similarity ratio we do the quotient of these two sides.

$\frac{AB}{DE}=\frac{15}{10}=\frac{3}{2}=1.5$

Answer

$1.5$

What is the difference between two similar triangles and congruent triangles?

The difference is that when two triangles are similar they have the same shape but their corresponding sides do not have to have equal sides, while when two triangles are congruent they have the same shape AND their corresponding sides are equal.

Exercise of similarity of triangles

Task

Demonstrate that the following triangles are similar

From the above we can observe that they have two pairs of equal angles

$\sphericalangle B= 45°= \sphericalangle E$

$\sphericalangle C= 75°= \sphericalangle F$

Then we say that the triangles are similar by the similarity criterion AA. They have the same shape but in different position.