on the same side, GHD, and the two interior angles on the same side, BGH, GHD, shall be together equal to two right angles.
For if the angle AGH be
not equal to the angle GHD,
one of them must be greater
than the other ; let the angle
AGH be the greater.
Then the angle AGH is greater
than the angle GHD ;
to each of them add the angle
BGH;
therefore the angles AGH, BGH are greater than the
angles BGH, GHD.
But the angles AGH, BGH are together equal to two
right angles ; [I. 13.
therefore the angles BGH, GHD are together less than
two right angles.
But if a straight line meet two straight lines, so as to
make the two interior angles on the same side of it, taken
together, less than two right angles, these straight lines
being continually produced, shall at length meet on that
side on which are the angles which are less than two
right angles. [Axiom 12.
Therefore the straight lines AB, CD, if continually pro-
duced, will meet.
But they never meet, since they are parallel by hypothesis.
Therefore the angle AGH is not unequal to the angle
GHD ; that is, it is equal to it.
But the angle AGH is equal to the angle EGB. [I. 15.
Therefore the angle EGB is equal to the angle GHD. [Ax. 1 .
Add to each of these the angle BGH.
Therefore the angles EGB, BGH are equal to the angles
BGH, GHD. [Axiom 2.
But the angles EGB, BGH are together equal to two
right angles. [I. 13.
Therefore the angles BGH, GHD are together equal to
two right angles. [Axiom 1.
Wherefore, if a straight line &c. q.e.d.
