Construction of Parallel Line

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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

Parallel Construction.png

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