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
Ailles rectangle between the two, with sides of lengths 2, 2 and (by the Pythagorean theorem) 2. The remaining space at the top of the rectangle is a right triangle with small angles of 15° and 75°. and sides of − 1, + 1 and 2. From the construction of the rectangle, it follows that and An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle, flanked by two 45°–45°–90° triangles in the lower corners. The lower-left triangle has sides of 1, 1, and , while the lower-right triangle has sides of , , and . The 30°–60°–90° triangle