Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/56

32
EUCLID'S ELEMENTS.

Let the straight line EF, which falls on the two straight lines AB, CD, make the exterior angle EGB equal to the interior and opposite angle GHD on the same side, or make the interior angles on the same side BGH, GHD together equal to two right angles : AB shall be parallel to CD.

Because the angle EGB is equal to the angle GHD, [I. 15.
and the angle EGB is also equal to the angle AGH, [1.15.
therefore the angle AGH is equal to the angle GHD;[Ax.l.
and they are alternate angles ;
therefore AB is parallel to CD. [I. 27.

Again; because the angles BGH, GHD are together equal to two right angles, [Hypothesis.
and the angles AGH, BGH are also together equal to two right angles, [1. 13.
therefore the angles AGH, BGH are equal to the angles BGH, GHD. Takeaway the common angle BGH; therefore the remaining angle AGH is equal to the remaining angle GHD; [Axiom 3. and they are alternate angles ; therefore AB is parallel to CD. [I. 27.

Wherefore, if a straight line &c. q.e.d.

PROPOSITION 29. THEOREM.

If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles.

Let the straight line EF fall on the two parallel straight lines AB, CD : the alternate angles AGH, GHD shall be equal to one another, and the exterior angle EGB shall be equal to the interior and opposite angle