Parallelism is Transitive

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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

Three Parallel Lines.png

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