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.
Take a point D at random on BC. Join AD. Construct the angle DAE equal to the angle ADC on the straight line DA and at the point A on it. Produce the straight line AF in a straight line with EA.
Since the straight line AD falling on the two straight lines BC and EF makes the alternate angles EAD and ADC equal to one another, therefore EAF is parallel to BC.
Therefore the straight line EAF has been drawn through the given point A parallel to the given straight line BC.
Incidentally, this construction also works in hyperbolic geometry, although different parallel lines through A are constructed for different points D.