Book 1
24
For, since the angle BAC is greater than the angle EDF, let there be constructed, on the straight line DE, and at the point D on it, the angle EDG equal to the angle BAC; [I. 23] let DG be made equal to either of the two straight lines AC, DF, and let EG, FG be joined.
24
Then, since AB is equal to DE, and AC to DG, the two sides BA, AC are equal to the two sides ED, DG, respectively; and the angle BAC is equal to the angle EDG; therefore the base BC is equal to the base EG. [I. 4]
24
Again, since DF is equal to DG, the angle DGF is also equal to the angle DFG; [I. 5] therefore the angle DFG is greater than the angle EGF.
24
Therefore the angle EFG is much greater than the angle EGF.
24
And, since EFG is a triangle having the angle EFG greater than the angle EGF, and the greater angle is subtended by the greater side, [I. 19] the side EG is also greater than EF.
24
But EG is equal to BC. Therefore BC is also greater than EF.
24
Therefore etc.
24
Q. E. D.
Proposition 25.
25
Enunciation If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other.
25
Proof. Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and AC to DF; and let the base BC be greater than the base EF;
25
I say that the angle BAC is also greater than the angle EDF.
25
For, if not, it is either equal to it or less.
25
Now the angle BAC is not equal to the angle EDF; for then the base BC would also have been equal to the base EF, [I. 4] but it is not; therefore the angle BAC is not equal to the angle EDF.
25
Neither again is the angle BAC less than the angle EDF; for then the base BC would also have been less than the base EF, [I. 24] but it is not; therefore the angle BAC is not less than the angle EDF.
25
But it was proved that it is not equal either; therefore the angle BAC is greater than the angle EDF.
25
Therefore etc.
25
Q. E. D.
Proposition 26.
