Through a given point to draw a straight line parallel to a given straight line.

Let $A$ be the given point, and $BC$ the given straight line;thus it is required to draw through the point $A$ a straight line parallel to the striaght line $BC$.Let a point $D$ be taken at random on $BC$, and let $AD$ be joined; on the straight line $DA$, and at the point $A$ on it, let the angle $DAE$ be constructed equal to the angle $ADC$ [Prop. 1.23]; and let the straight line $AF$ be produced in a straight line with $EA$.Then, since the straight line $AD$ falling on the two straight lines $BC$, $EF$ has made the alternate angles $EAD$, $ADC$ equal to one another,therefore $EAF$ is parallel to $BC$.Therefore through the given point $A$ the straight line $EAF$ has been drawn parallel to the given straight line $BC$.Q.E.F.