13.1 The definition of triangle.

A triangle is one of the basic shapes in geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A,B and C is denoted . In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimentional Euclidean space. Triangles are assumed to be two-dimensional figures, unless the context provides othervise.

13.2 Basic facts.

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements, around 300 BC.

1)      The measures of the interior angles of a triangle in Euclidean space always add up to 180 degrees. This allows determination of the measure of the third angle of any triangle given the measure of two angles.

2)      The exterior angle theorem: An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two angles that are not adjacent to it. The sum of the measures of the exterior angles (one for each vertex) of any triangle is 360 degrees.

13.3 Types of triangle – by relative lengths of sides.

Picture: Euler diagram of types of triangle, using definition that isosceles triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles.

Triangles can be classified according to the relative lengths of their sides:

1)      EQUILATERAL TRIANGLE - In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60˚.

2)      ISOSCALES TRIANGLE – In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the isosceles triangle theorem, which was known by Euclid. Some mathematicians define an isosceles triangle to have exactly two equal sides, wheres others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles.  

3)      SCALENE TRIANGLE – In a scalene triangle, all sides are unequal, and equivalently all angles are unequal. Right triangles are scalene if and only if not isosceles.

In diagrams representing triangles (and other geometric figures),

• “tick” marks along the sides are used to denote sides of equal lengths – the equilateral triangle has tick marks on all 3 sides, the isosceles on 2 sides. The scalene has single, double and triple tick marks, indicating that no sides are equal.
• Similarly, arcs on the inside of the vertices are used to indicate equal angles. The equilateral triangle indicates all 3 angles are equal; the isosceles shows 2 identical angles. The scalene indicates by 1, 2, and 3 arcs that no angles are equal.

13.4 Types of triangle – by internal angles.

Triangles can also be classified according to their internal angles, measured here in degrees.

1)      RIGHT TRIANGLE – A right triangle (or right-angled triangle, formerly called a rectangle triangle) has one of its interior angles measuring 90˚ (a right angle). The side opposite to the right angle is the hypothenuse; it is the longest side of the right triangle. The other two sides are called legs or catheti (singular: cathetus)  of the triangle. Right triangles obey the Pythagorean theorem. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3-4-5 right triangle, where 3² + 4² = 5² In this situation, 3, 4, and 5 are Pythagorean Triple. The other one is an isosceles triangle that has 2 angles that each measure 45 degrees.

2)      OBLIQUE TRIANGLE – Triangles that do not have an angle that measures 90˚ are called oblique triangles.

3)      ACUTE TRIANGLE – A triangle that has all interior angles measuring less than 90˚ is an acute triangle or acute-angled triangle. If the greatest side length is c, then: a² + b² > c²

4)      OBTUSE TRIANGLE – A triangle that has one interior angle that measures more than 90˚ is an obtuse triangle or obtuse-angled triangle. If the greatest side length is c, then a² + b² < c².

5)      DEGENERATE – A “triangle” with an interior angle of 180˚ (and collinear vertices).