Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/58

34 PROPOSITION 30. THEOREM.

Straight lines which are parallel to the same straight line are parallel to each other.

Let AB, CD be each of them parallel to EF: AB shall be parallel to CD.

Let the straight line GHK cut AB, EF, CD.

Then, because GHK cuts the parallel straight lines AB, EF, the angle AGH is equal to the angle GHF. [I. 29. Again, because GK cuts the parallel straight lines EF, CD, the angle GHF is equal to the angle GKD. [I. 29. And it was shewn that the angle AGK is equal to the angle GHF. Therefore the angle AGK is equal to the angle GKD;[Ax. 1. and they are alternate angles ; therefore AB is parallel to CD.

Wherefore, straight lines &c.

PROPOSITION 31. PROBLEM.

To draw a straight line through a given point parallel to a given straight line.

Let A be the given point, and BC the given straight line : it is required to draw a straight line through the point A parallel to the straight line BC.

In BC take any point D, and join AD ; at the point A in the straight line AD, make the angle DAE equal to the angle ADC, [I.23. and produce the straight line EA to F. EF shall be parallel to BC.