Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/57

Rh 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.