Special right triangle is 90°, leaving the remaining angle to be 30°. Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that cannot all be rational numbers of degrees. (This follows from Niven's theorem.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio where "m" and "n" are any positive integers such that . There are several Pythagorean triples which are well-known, including those with sides in the
Special right triangle : , which follows immediately from the Pythagorean theorem. Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely . Of all right triangles, the 45°–45°–90° degree triangle has the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely . Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles. This is a triangle whose three angles