(related to Proposition: 1.30: Transitivity of Parallel Lines)

- For let the straight line $GK$ fall across ($AB$, $CD$, and $EF$).
- And since the straight line $GK$ has fallen across the parallel straight lines $AB$ and $EF$, (angle) $AGK$ (is) thus equal to $GHF$ [Prop. 1.29].
- Again, since the straight line $GK$ has fallen across the parallel straight lines $EF$ and $CD$, (angle) $GHF$ is equal to $GKD$ [Prop. 1.29].
- But $AGK$ was also shown (to be) equal to $GHF$.
- Thus, $AGK$ is also equal to $GKD$.
- And they are alternate (angles).
- Thus, $AB$ is parallel to $CD$ [Prop. 1.27].
- Thus, (straight lines) parallel to the same straight line are also parallel to one another.
- (Which is) the very thing it was required to show.∎

**Fitzpatrick, Richard**: Euclid's "Elements of Geometry"