Definition:Triangle (Geometry)
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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.
Parts of a Triangle
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.
Apex
Having chose one side as the base, the opposite angle is called the apex.
Height
The height of a triangle is the length of a perpendicular from the apex to the base.
Otherwise known as the altitude.
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 $\angle A$.
Types of Triangle
Isosceles Triangle
An isosceles triangle is a triangle in which two sides are the same length and the third side is a different length.
Equilateral Triangle
An equilateral triangle is a triangle in which all three sides are the same length:
Thus it can be seen that an equilateral triangle is a special case of an isosceles triangle.
Scalene Triangle
A scalene triangle is a triangle in which all three sides are different lengths.
Right-Angled Triangle
A right-angled triangle, or right triangle is a triangle in which one of the angles 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 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.
Also see
Euclid's Definition
As Euclid defined it:
- 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.
(The Elements: Book I: Definition $19$)
- Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
(The Elements: Book I: Definition $20$)
- Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
