Construction of a Parallel

Theorem

Given an infinite straight line, and a given point not on that straight line, it is possible to draw a parallel to the given straight line.

Construction

Let $A$ be the point, and let $BC$ be the infinite straight line.

We take a point $D$ at random on $BC$, and construct the segment $AD$.

We construct $\angle DAE$ equal to $\angle ADC$ on $AD$ at point $A$.

We extend $AE$ into an infinite straight line.

Then the line $AE$ is parallel to the given infinite straight line $BC$ through the given point $A$.

Proof

Since the transversal $AD$ cuts the lines $BC$ and $AE$ and makes $\angle DAE = \angle ADC$, it follows that $EA \parallel BC$.

$\blacksquare$

Historical Note

This is Proposition 31 of Book I of Euclid's The Elements.