The reason basically comes down to the fact that every time you perform a compass and straight-edge construction, you are constructing quadratic extensions of number fields, whereas trisection (in general) needs a cubic extension.
The 'in general' part is important: of course, you can trisect specific angles like a 90 degree angle.
(The construction of number field extensions is happening because let's say you construct a 90 degree angle, well that allows you to construct an isosceles triangle with equal sides length 1, and therefore you construct the square root of 2, so the extension Q(sqrt(2))
The 'in general' part is important: of course, you can trisect specific angles like a 90 degree angle.
(The construction of number field extensions is happening because let's say you construct a 90 degree angle, well that allows you to construct an isosceles triangle with equal sides length 1, and therefore you construct the square root of 2, so the extension Q(sqrt(2))