Two points must be made. Firstly, some angles are trisectable with a ruler and a compass. An obvious example is supplied by the three quarters of the straight angle - 67.5 o . Therefore, the latter also admits a classical trisection. However, and this is where the difficulty lies, an arbitrary angle can't be trisected with a ruler and a compass, as stated above.
Octagons are constructible on the heels of squares with a single angle bisection.
All polygons obtained from the above four by doubling the number of sides are also constructible. Not so a heptagon, a 7-sided polygon. As Gauss showed, the heptadecagon was only a particular case of a family of constructible regular polygons. Wantzel later proved that no other regular polygon has this property. Each depends on showing that roots of certain third degree polynomial equations are not constructible.