Book 1
29
Let the angle BGH be added to each; therefore the angles AGH, BGH are greater than the angles BGH, GHD.
29
But the angles AGH, BGH are equal to two right angles; [I. 13] therefore the angles BGH, GHD are less than two right angles.
29
But straight lines produced indefinitely from angles less than two right angles meet; [Post. 5] therefore AB, CD, if produced indefinitely, will meet; but they do not meet, because they are by hypothesis parallel.
29
Therefore the angle AGH is not unequal to the angle GHD, and is therefore equal to it.
29
Again, the angle AGH is equal to the angle EGB; [I. 15] therefore the angle EGB is also equal to the angle GHD. [C.N. 1]
29
Let the angle BGH be added to each; therefore the angles EGB, BGH are equal to the angles BGH, GHD. [C.N. 2]
29
But the angles EGB, BGH are equal to two right angles; [I. 13] therefore the angles BGH, GHD are also equal to two right angles.
29
Therefore etc.
29
Q. E. D.
Proposition 30.
30
Enunciation Straight lines parallel to the same straight line are also parallel to one another.
30
Proof. Let each of the straight lines AB, CD be parallel to EF; I say that AB is also parallel to CD.
30
For let the straight line GK fall upon them;
30
Then, since the straight line GK has fallen on the parallel straight lines AB, EF, the angle AGK is equal to the angle GHF. [I. 29]
30
Again, since the straight line GK has fallen on the parallel straight lines EF, CD, the angle GHF is equal to the angle GKD. [I. 29]
30
But the angle AGK was also proved equal to the angle GHF; therefore the angle AGK is also equal to the angle GKD; [C.N. 1] and they are alternate.
30
Therefore AB is parallel to CD.
30
Q. E. D.
Proposition 31.
31
Enunciation Through a given point to draw a straight line parallel to a given straight line.
31
Proof. Let A be the given point, and BC the given straight line; thus it is required to draw through the point A a straight line parallel to the straight line BC.
