# 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.

In the words of Euclid:

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

(*The Elements*: Book $\text{I}$: Proposition $31$)

## 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 Interior Angles implies Parallel Lines it follows that $EA \parallel BC$.

$\blacksquare$

## Historical Note

This theorem is Proposition $31$ of Book $\text{I}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 1*(2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions