Book 11
22
And, since the angles ABC, GHK are greater than the angle DEF, while the angle ABC is equal to the angle KHL, therefore the angle GHL is greater than the angle DEF.
22
And, since the two sides GH, HL are equal to the two sides DE, EF, and the angle GHL is greater than the angle DEF, therefore the base GL is greater than the base DF. [I. 24]
22
But GK, KL are greater than GL.
22
Therefore GK, KL are much greater than DF.
22
But KL is equal to AC; therefore AC, GK are greater than the remaining straight line DF.
22
Similarly we can prove that AC, DF are greater than GK, and further DF, GK are greater than AC.
22
Therefore it is possible to construct a triangle out of straight lines equal to AC, DF, GK. Q. E. D.
PROPOSITION 23.
23
To construct a solid angle out of three plane angles two of which, taken together in any manner, are greater than the remaining one: thus the three angles must be less than four right angles.
23
Let the angles ABC, DEF, GHK be the three given plane angles, and let two of these, taken together in any manner, be greater than the remaining one, while, further, the three are less than four right angles; thus it is required to construct a solid angle out of angles equal to the angles ABC, DEF, GHK.
23
Let AB, BC, DE, EF, GH, HK be cut off equal to one another, and let AC, DF, GK be joined; it is therefore possible to construct a triangle out of straight lines equal to AC, DF, GK. [XI. 22]
