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. q.e.d.
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.
