More Grade School Triangles

Planar triangles with each side having a positive integer length are among the simplest geometrical objects imaginable. What can be said about the angles of such triangles? In particular, are these angles rational or at least algebraic numbers when measured in degrees? This article demonstrates that the answer in general is negative, except for three distinguished families of triangles. One family is well known since antiquity: With the largest angle equal to 90 degrees, it simply is the family of Pythagorean right triangles. Though not nearly as well-known, the other two families also deserve to be part of every geometry teacher’s toolkit.