Classifying Triangles

A triangle is the smallest closed shape you can make with straight sides. It has three corners, called vertices, three sides and three angles, and once you label the corners A, B and C you can talk about any part of it precisely. The sides are AB, BC and CA, and the angles are angle A, angle B and angle C. Notice that each side sits opposite one corner, so side AB is the side opposite angle C. Triangles are sorted in two completely separate ways: one looks only at the sides, and the other looks only at the angles. Every triangle picks up one name from each list, so most triangles have two names.

The first way to sort a triangle is by counting how many of its sides are the same length, and there are only three possible answers. If all three sides are equal the triangle is equilateral, and its diagram carries a tick on every side. If exactly two sides are equal it is isosceles, and the two matching sides carry a tick each. If no two sides are equal it is scalene, and there are no matching ticks anywhere. Three, two or none, and that covers every triangle there is. You never need a ruler for this in a test, because the tick marks on the diagram already tell you which sides match.

The second way to sort a triangle is by its angles, and the quickest route is to find the largest angle and look at that one alone. If the largest angle is still under ninety degrees then every angle is under ninety, and the triangle is acute. If the largest angle is exactly ninety degrees the triangle is right-angled, and the diagram shows that corner with a small square. If the largest angle is bigger than ninety degrees the triangle is obtuse. One angle decides it every time, so you never have to check all three. The triangles here measure sixty-five, forty-five and seventy; ninety, thirty-five and fifty-five; and fifty, thirty and one hundred.

Because the two lists are separate, a full description uses one name from each. A triangle with angles of ninety, forty-five and forty-five is right-angled from the angle list and isosceles from the side list, so you call it a right-angled isosceles triangle. A triangle with three sixty degree angles is acute and equilateral. A triangle measuring one hundred, fifty and thirty is obtuse and scalene. So ask yourself two questions every time: how many sides are equal, and how big is the largest angle. The two answers give you the two halves of the name, and they never contradict each other.

The two lists are not really independent, because equal sides always sit opposite equal angles. In this isosceles triangle the two ticked sides are equal, and the two angles opposite them, marked with arcs, are equal as well. That is why an isosceles triangle is often drawn with matching arcs at the two base corners. The rule also runs backwards: if you spot two equal angles in a triangle, then the sides opposite them must be equal, so the triangle is isosceles. An equilateral triangle takes this to the end. Three equal sides give three equal angles, and each one of them turns out to be sixty degrees.