# Definition:Triangle (Geometry)

## Definition

A triangle is a polygon with exactly three sides.

Thus a triangle is a $2$-simplex.

Because it is a polygon, it follows that it also has three vertices and three angles.

## Parts of a Triangle

The two sides of a triangle which form a particular vertex are referred to as adjacent to that angle.

Similarly, the two vertices of a triangle to which a particular side contributes are referred to as adjacent to that side.

### Opposite

The side of a triangle which is not one of the sides adjacent to a particular vertex is referred to as its opposite.

Thus, each vertex has an opposite side, and each side has an opposite vertex.

### Base

For a given triangle, one of the sides can be distinguished as being the base.

It is immaterial which is so chosen.

The usual practice is that the triangle is drawn so that the base is made horizontal, and at the bottom.

In the above diagram, it would be conventional for the side $AC$ to be identified as the base.

### Apex

Having selected one side of a triangle to be the base, the opposite vertex to that base is called the apex.

In the above diagram, if $AC$ is taken to be the base of $\triangle ABC$, then $B$ is the apex.

### Height

The height of a triangle is the length of a perpendicular from the apex to the base.

## Conventional Nomenclature

The vertices of a triangle are conventionally labeled $A, B, C$ (or with other uppercase letters), and the sides with lowercase letters corresponding to the opposite vertex, as above.

In order to emphasize that a particular vertex being referred to is in fact a vertex, the symbol $\angle$ is often placed by the letter corresponding to that vertex.

Thus, for example:

$\angle A$ is adjacent to sides $b$ and $c$
Side $a$ is adjacent to $\angle B$ and $\angle C$
$\angle A$ is opposite side $a$
Side $a$ is opposite $\angle A$.

## Types of Triangle

### Isosceles Triangle

An isosceles triangle is a triangle in which two sides are the same length.

### Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are the same length:

That is, a regular polygon with $3$ sides.

### Scalene Triangle

A scalene triangle is a triangle in which all three sides are of different lengths.

### Right-Angled Triangle

A right-angled triangle is a triangle in which one of the vertices is a right angle.

Note that in order to emphasise the nature of the right angle in such a triangle, a small square is usually drawn inside it.

### Oblique Triangle

An oblique triangle is a triangle in which none of the vertices are right angles.

### Acute Triangle

An acute triangle is a triangle in which all three of the vertices are acute angles.

### Obtuse Triangle

An obtuse triangle is a triangle in which one of the vertices is an obtuse angle.

## Also known as

The word trigon can occasionally be seen, but this is rare.

## Technical Note

The $\LaTeX$ code to generate $\triangle ABC$ is written \triangle ABC.

Note that \Delta is not to be used, as, although producing a symbol similar in shape, this is actually the uppercase Greek letter delta: $\Delta$

## Euclid's Definition

In the words of Euclid:

Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multi-lateral those contained by more than four straight lines.