Any solid angle is contained by plane angles less than four right angles.
Proof
Cut a small polygon by a plane near the apex; the sum of the
exterior angles of this polygon is less than 4⋅90∘ (by
I.32 / I.34 applied to the polygon). The interior face-angles at
the apex are the supplements of these exterior angles, so their sum
falls strictly short of 4⋅90∘.