Bit of a tautology, I suppose. The pretty statement comes from the observation that an angle $\theta$ is constructible (as an angle in a right triangle, for example) if and only if $\sin \theta$ or $\cos \theta$ or $\tan \theta$ is constructible, these three conditions being equivalent for, say, acute angles. It comes more or less for free that the sum or difference of constructible angles is also constructible.
The attractive part is this: the constructible angles on the surface of the unit sphere, and the constructible angles in the hyperbolic plane of curvature $-1,$ are exactly the same as the constructible angles in the traditional Euclidean plane.
The spell checker prefers constructable. i thought it was i.
Just for flavour, or flavor, it is generally impossible in the hyperbolic plane to trisect a line segment, which seems mysterious. However, if you put a line segment on the equator of the unit sphere and consider the triangle made with the North Pole, you see that you are asked to trisect the angle at the North pole. And angle trisection is often impossible; we know about that from the Euclidean plane.