# Similarities among figures: triangles

Colloquially, it is said that two objects are similar if they have the same shape but a different size. Notice that both objects can have different orientations but still resemble one another, meaning that shape is the only important thing when determinig if two objects are similar or not.

We define a similarity as the composition of a rotation or symmetry and a movement in the plane. To learn more about this type of geometric transformation, there is a unit dedicated to it.

When it comes to triangles, we will say that two triangles are similar if they have the same shape. The shape depends on the angles of the triangle (it is not like this in a rectangle's case, for example, where all the angles are straight but which shape can be more or less lengthened, that is to say that length / width depends on the quotient).

It is possible to simplify, in this way, the definition: two triangles are similar if their angles are equal two by two.

In the diagram, the corresponding angles are $A = A'$, $B = B'$ and $C = C'$. To denote that two triangles $ABC$ and $DEF$ are similar we write $ABC \sim DEF$, where the order indicates the correspondence between the angles: $A, B$ and $C$ correspond with $D, E$ and $F$, respectively.

A similarity has the property (that characterizes it) of multiplying all the lengths by the same factor. Hence the reason why length image / length origin are equal, which leads us to the second characterization of the similar triangles: Two triangles are similar if the reasons of the corresponding sides are equal.

From the two previous characterizations, we can extract the following equation: $$(ABC \sim A'B'C') \Leftrightarrow \begin{array}{c} \widehat{A}=\widehat{A'} \\ \widehat{B}=\widehat{B'} \\ \widehat{C}=\widehat{C'} \end{array} \Leftrightarrow \dfrac{\overline{A'B'}}{\overline{AB}}=\dfrac{\overline{A'C'}}{\overline{AC}}=\dfrac{\overline{B'C'}}{\overline{BC}}$$

Notice that with the notation $\widehat{A}$ we refer to the angle that is in the apex $A$.

From these equalities, two important results can be extracted:

1. All the equilateral triangles are similar.
2. If two triangles have two equal angles, the third ones are also equal.

Finally, we are going to give a few basic properties of the similarity of the triangles:

• Reflexive property: Any triangle is similar to itself.

• Symmetrical property: If a triangle is similar to another, that one is similar to the first one.

• Transitive property: If a triangle is similar to another, and at the same time this one is similar to a third one, the first one is similar to the third one.

Given the triangle $ABC$ with sides $a = 5, b = 5$ and $c = 10$ and the $A'B'C'$ with sides $a' = 8, b' = 8$ and $c' = 10$, we can easily see that these two triangles cannot be similar since the quotient of the lengths is different: So, we have: $$\dfrac{a}{a'}=\dfrac{b}{b'}=\dfrac{c}{c'}\Rightarrow \dfrac{5}{8}=\dfrac{5}{8}\neq\dfrac{10}{10}=1$$

Therefore, these two triangles are not similar.

## Criteria for similarity of triangles

### Rectangular triangles

In this first part we are going to give a criteria of similarities concerning rectangular triangles.

• 1) Two rectangular triangles are similar if they have an equal acute angle.

$\widehat{C}=\widehat{C'}$

• 2) Two rectangular triangles are similar if they have two proportional legs.

$\dfrac{b}{b'}=\dfrac{c}{c'}$

For example, a rectangular triangle with legs $a = 3$ and $b = 4$, and another one with legs $a' = 6$ and $b' = 8$ are similar since the ratios between the leg two are equal, so,$$\dfrac{3}{4}=\dfrac{3\cdot2}{4\cdot2}=\dfrac{6}{8}$$

• 3) Two rectangular triangles are similar if their hypotenuse and their leg are proportional.

$\dfrac{a}{a'}=\dfrac{b}{b'}$

Let $ABC$ be a rectangular triangle with a leg length of $a = 4$ and a hypotenuse of $b = 5$. On the other hand, let $A'B'C'$ be another rectangular triangle with a leg length $a' = 16$ and a hypotenuse $b' = 20$. As such, both triangles are similar since the reasons between the two legs and the two hypotenuses match up. So, $$\dfrac{4}{5}=\dfrac{4\cdot4}{5\cdot4}=\dfrac{16}{20}$$

### Triangles in general

In this second part, we will give more general criteria to determine the similarities between triangles.

• 4) Two triangles are similar if they have two equal angles.

$A=A' \ \ \ B=B'$

• 5) Two triangles are similar if they have proportional sides.

$\dfrac{a}{a'}=\dfrac{b}{b'}=\dfrac{c}{c'}$

• 6) Two triangles are similar if they have two proportional sides and the angle between them is equal.

$B=B' \ \ \ \dfrac{a}{a'}=\dfrac{c}{c'}$

Let $ABC$ be a triangle that has sides $a=5$ and $b=7$ and the angle between them is $35^\circ$. Let $A'B'C'$ be another triangle with sides $a=2$ and $b=3$ where the angle between them is $35^\circ$. Then, although both two angles match up, the triangles are not similar since the reasons between these two sides are not equal: $$\dfrac{5}{2}\neq\dfrac{7}{3}$$ dado que $$5\cdot3=15\neq14=7\cdot2$$