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.*

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.*

Q.E.F.

Incidentally, this construction also works in hyperbolic geometry, although different parallel lines through *A* are constructed for different points *D.*