# Classification and properties of the triangles

## Classification of triangles

Triangles can be classified according to a different criteria:

• According to its sides
• According to its angles

### Classification of triangles depending on the sides

#### Equilateral triangle

If its three sides have the same length (three internal angles measuring $60$ degrees).

#### Isosceles triangle

If it has two sides of the same length. The angles that are opposed to these sides have the same measurement.

#### Scalene triangle

If all its sides have different lengths. In a scalene triangle there are no angles with the same measurement.

### Classification of triangles depending on the angles

#### Triangle Rectangle:

If it has a right interior angle $(90^\circ)$. Both sides conforming to the right angle are named leg and hypotenuse.

#### Obtuse triangle

If one of its angles is obtuse (higher than $90^\circ$); the other two are acute (less than $90^\circ$).

#### Acute angled triangle

When its three angles are less than $90^\circ$; the equilateral triangle is a particular case of acute angled triangle.

#### Equiangular triangle

Normally it is called an equilateral triangle and it has already been commented on previously.

## Properties of the triangles

Triangles Equilateral Isosceles Scalen
Acute angled

Rectangle

Obstuse angled

We can see in the previous scheme that the classifications commented on in the previous paragraph can be combined in pairs (one from every paragraph).

Thus, we have the following characteristics:

• Isosceles acute angled triangle: All the angles are acute, two of them being equal, and the other different from these two. This triangle is symmetrical regarding the height, making it different from the others.
• Acute angled scalene triangle: All its angles are acute and all are different, it does not have symmetry axes.

The rectangular triangles can be:

• Rectangular isosceles Triangle: with one right angle and two equal acute ones (of $45^\circ$ each one); two sides are equal and the other is different, naturally the equal sides are the legs, and the different one is the hypotenuse; it is symmetrical regarding the height that connects the right angle up to the hypotenuse.
• Rectangular scalene triangle: it has a right angle and all its sides and angles are different.

The obtuse triangles are:

• Isosceles obtuse triangle: it has an obtuse angle, and two equal sides that are those that begin in the obtuse angle, the other side is bigger than the other two.
• Obtuse scalene triangle: it has an obtuse angle and all its sides are different.