The basic elements of any triangle are its sides and vertices. Triangles are classified depending on relative sizes of their elements.

As regard their sides, triangles may be

• Scalene
• Isosceles
• Equilateral

And as regard their angles, triangles may be

• Acute
• Right
• Obtuse
• Equiangular

A triangle is scalene if all of its three sides are different. If two of its sides are equal, a triangle is called isosceles. A triangle with all three equal sides is called equilateral. S. Schwartzman's The Words of Mathematics explain the etymology (the origins) of the words. The first two are of Greek (and related) origins; the word "equilateral" is of Latin origin:

This is how the two approaches are distinguished with Venn diagrams:

As regard the angles, a triangle is equiangular is all three of its angles are equal. Very early in the Elements (I.5 and I.6) Euclid showed that in an isosceles triangle the base angles are equal and, conversely, the sides opposite equal angles are equal. From here, for a triangle, the properties of being equilateral and equiangular are equivalent, and the latter is seldom mentioned. (For a polygon with the number of sides greater than 3 the equivalence no longer holds.)

In Euclidean geometry, the sum of the angles in a triangle equals 180^o. It follows that a triangle may have at most one obtuse or even right angle. (This also follows from the Exterior Angle Theorem.) If one of the angles in a triangle is obtuse, the triangle is called obtuse. A triangle with one right angle is right. Otherwise, a triangle is acute; for all of its angles are acute. (All the definitions are naturally exclusive. There is no possible ambiguity.)

References

1. S. Schwartzman, The Words of Mathematics, MAA, 1994

Copyright © 1996-2008 Alexander Bogomolny