Construction of Parallel Line

Theorem
Given a 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.

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

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

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
The transversal $AD$ cuts the lines $BC$ and $AE$ and makes $\angle DAE = \angle ADC$.

From Equal Alternate Angles implies Parallel Lines it follows that $EA \parallel BC$.