# Parallelism is Transitive

## Contents

## Theorem

Parallelism between straight lines is a transitive relation.

In the words of Euclid:

*Straight lines parallel to the same straight line are also parallel to one other.*

(*The Elements*: Book $\text{I}$: Proposition $30$)

## Proof

Let the straight lines $AB$ and $CD$ both be parallel to the straight line $EF$.

Let the straight line $GK$ be a transversal that cuts the parallel lines $AB$ and $EF$.

By Parallelism implies Equal Alternate Interior Angles:

- $\angle AGK = \angle GHF$

By Playfair's Axiom, there is only one line that passes through $H$ that is parallel to $CD$ (namely $EF$).

Therefore the transversal $GK$ cannot be parallel to $CD$ and the two lines must therefore intersect.

The straight line $GK$ also cuts the parallel lines $EF$ and $CD$.

So from Parallelism implies Equal Corresponding Angles:

- $\angle GHF = \angle GKD$.

Thus $\angle AGK = \angle GKD$.

So from Equal Alternate Interior Angles implies Parallel Lines:

- $AB \parallel CD$

$\blacksquare$

## Historical Note

This theorem is Proposition $30$ of Book $\text{I}$ of Euclid's *The Elements*.

Note that while this result applies to all parallel lines in Euclidean geometry, this proof is only valid when all three lines are in the same plane.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 1*(2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions - 1968: M.N. Aref and William Wernick:
*Problems & Solutions in Euclidean Geometry*... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.7$