acute, right and obtuse triangles

The other natural way to sort triangles is by their largest angle. An acute triangle has all three angles less than 90 degrees. A right triangle has exactly one 90-degree angle. An obtuse triangle has one angle greater than 90 degrees. Because the three angles add to 180, a triangle can have at most one right or obtuse angle, so the largest angle alone settles the type.

Precisely: classify by max(angle A, angle B, angle C). If that maximum is below 90, the triangle is acute; if it equals 90, the triangle is right and the side opposite the right angle is the hypotenuse (the longest side); if it exceeds 90, the triangle is obtuse. A right triangle is the one where the Pythagorean relation a^2 + b^2 = c^2 holds, with c the hypotenuse, which is why right triangles anchor so much of geometry and trigonometry. There is no such thing as a triangle with two right angles or two obtuse angles — their two angles alone would already reach or exceed 180.

Keep this angle-based scheme separate from the side-based one (scalene, isosceles, equilateral); a triangle carries one label from each. For example, an equilateral triangle is always acute (all angles 60), while a right triangle may be either scalene or isosceles (the 45-45-90 triangle is an isosceles right triangle).