*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