Lines Parallel to Same Line not in Same Plane are Parallel to each other

Theorem

In the words of Euclid:

Straight lines which are parallel to the same straight line and are not in the same plane with it are also parallel to one another.

(The Elements: Book $\text{XI}$: Proposition $9$)

Proof

Let $AB$ and $CD$ be straight lines which are both parallel to another straight line $EF$ which is in a different plane.

It is to be demonstrated that $AB$ is parallel to $CD$.

Let $G$ be an arbitrary point on $EF$.

Let $GH$ be drawn at right angles to $EF$, in the plane holding both $AB$ and $EF$.

Let $GK$ be drawn at right angles to $EF$, in the plane holding both $CD$ and $EF$.

We have that $EF$ is at right angles to each of the straight lines $GH$ and $GK$.

Therefore from Proposition $4$ of Book $\text{XI} $: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane:

$EF$ is at right angles to the plane through $GH$ and $GK$.

We have that $EF$ is parallel to $AB$.

Therefore from Proposition $8$ of Book $\text{XI} $: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane:

$AB$ is at right angles to the plane through $GH$ and $GK$.

For the same reason:

$CD$ is at right angles to the plane through $GH$ and $GK$.

Therefore each of the straight lines $AB$ and $CD$ is at right angles to the plane through $GH$ and $GK$.

From Proposition $6$ of Book $\text{XI} $: Two Lines Perpendicular to Same Plane are Parallel:

$AB$ is parallel to $CD$.

$\blacksquare$

Historical Note

This proof is Proposition $9$ of Book $\text{XI}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions