Definition:Triangle (Geometry)

Definition
A triangle (or, rarely, trigon) is a polygon with three sides.

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

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

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

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

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

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.

Conventional Nomenclature


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

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

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 $$\angle A$$.

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


 * EquilateralTriangle.png

Isosceles Triangle
An isosceles triangle is a triangle in which two sides are the same length and the third side is a different length.


 * IsoscelesTriangle.png

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


 * ScaleneTriangle.png

Right-Angled Triangle
A right-angled triangle, or right triangle is a triangle in which one of the angles is a right angle.


 * RightTriangle.png

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

Note the following nomenclature in the context of a right triangle:
 * The opposite side to the right angle is known as the hypotenuse.
 * For a given non-right angle, the adjacent side which is not the hypotenuse is referred to as the adjacent.

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

Acute-Angled Triangle
An acute-angled triangle, or an acute triangle, is a triangle in which all three of the angles are acute angles.

Obtuse-Angled Triangle
An obtuse-angled triangle, or an obtuse triangle, is a triangle in which one of the angles is a obtuse angle.

Euclid's Definition
Euclid, in his definitions at the start of his Book I, refers to a trilateral figure, that is, a rectilineal figure which has three sides.