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

Proof

 * Euclid-XI-9.png

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

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

Therefore from :
 * $AB$ 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 :
 * $AB$ is parallel to $CD$.