Any solid angle is contained by plane angles whose sum is less than four right angles.
Let the angle at A be a solid angle contained by the plane angles BAC , CAD , and DAB . I say that the sum of the angles BAC , CAD , and DAB is less than four right angles. Take points B , C , and D at random on the straight lines AB , AC , and AD respectively, and join BC , CD , and DB . Now, since the solid angle at B is contained by the three plane angles CBA , ABD , and CBD , and the sum of any two is greater than the remaining one, therefore the sum of the angles CBA and ABD is greater than the angle CBD . For the same reason the sum of the angles BCA and ACD is greater than the angle BCD , and the sum of the angles CDA and ADB is greater than the angle CDB .