Construction of Parallel Line

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 $EA \parallel BC$.