Straight lines parallel to the same straight line are also parallel to one another.
[Desmos graphs can be found here]
If the two lines AB and CD are both parallel to another line EF, we will prove that AB and CD are parallel to each other.
Let the straight line GK fall upon the three lines as shown.
By Proposition 29 (alternate angles are equal), since AB and EF are parallel angle AGK is equal to the angle GHF.
Since EF is parallel to CD, Proposition 29 tells us that angle GHF is equal to angle GKD. (Exterior angle is equal to the interior and opposite angle.)
Therefore angle AGK is equal to angle GKD. (AGK = GHF = GKD)
Since angles AGK and GKD are alternate, Proposition 27 tells us that the lines AB and CD are parallel to one another. (If a straight line falls on two straight lines and the alternate angles are equal, then the two straight lines are parallel.)
Q.E.D.
The next proposition involves the construction of a line through a given point that is parallel to a given line – George