Any solid angle is contained by plane angles less than four right angles.
任意立体角被小于四个直角的平面角所包含。
Let the angle at A be a solid angle contained by the plane angles BAC, CAD, DAB; I say that the angles BAC, CAD, DAB are less than four right angles. For let points B, C, D be taken at random on the straight lines AB, AC, AD respectively, and let BC, CD, DB be joined.
设A为立体角顶点，平面角BAC、CAD、DAB构成该立体角。在AB、AC、AD上分别任取点B、C、D，连接BC、CD、DB。
Now, since the solid angle at B is contained by the three plane angles CBA, ABD, CBD, any two are greater than the remaining one; [XI. 20] therefore the angles CBA, ABD are greater than the angle CBD.
由于B处的立体角由三个平面角CBA、ABD、CBD构成，其中任意两个之和大于第三个，故角CBA与ABD之和大于角CBD。同理，角BCA与ACD之和大于角BCD，角CDA与ADB之和大于角CDB。
For the same reason the angles BCA, ACD are also greater than the angle BCD, and the angles CDA, ADB are greater than the angle CDB; therefore the six angles CBA, ABD, BCA, ACD, CDA, ADB are greater than the three angles CBD, BCD, CDB. But the three angles CBD, BDC, BCD are equal to two right angles; [I. 32] therefore the six angles CBA, ABD, BCA, ACD, CDA, ADB are greater than two right angles.
因此，六个角CBA、ABD、BCA、ACD、CDA、ADB之和大于三个角CBD、BCD、CDB之和。而三角形BCD的内角和等于两直角，故这六个角之和大于两直角。
And, since the three angles of each of the triangles ABC, ACD, ADB are equal to two right angles, therefore the nine angles of the three triangles, the angles CBA, ACB, BAC, ACD, CDA, CAD, ADB, DBA, BAD are equal to six right angles; and of them the six angles ABC, BCA, ACD, CDA, ADB, DBA are greater than two right angles; therefore the remaining three angles BAC, CAD, DAB containing the solid angle are less than four right angles.
三角形ABC、ACD、ADB的内角和各等于两直角，故三个三角形的九个角之和等于六直角。其中六个角ABC、BCA、ACD、CDA、ADB、DBA之和大于两直角，因此剩下的三个角BAC、CAD、DAB之和小于四直角。