# Lines Joining Equal and Parallel Straight Lines are Parallel

## Theorem

The straight lines joining equal and parallel straight lines at their endpoints, in the same direction, are themselves equal and parallel.

In the words of Euclid:

*The straight lines joining equal and parallel straight lines (at the extremities which are) in the same direction (respectively) are themselves equal and parallel.*

(*The Elements*: Book $\text{I}$: Proposition $33$)

## Proof

Let $AB, CD$ be equal and parallel.

Let $AC, BD$ join their endpoints in the same direction.

Draw $BC$.

From Parallelism implies Equal Alternate Angles:

- $\angle ABC = \angle BCD$

We have that $AB, BC$ are equal to $DC, CB$ and $\angle ABC = \angle BCD$.

It follows from Triangle Side-Angle-Side Equality that $AC = BD$.

Also, $\triangle ABC = \triangle DCB$, and thus $\angle ACB = \angle CBD$.

We have that $BC$ falling on the two straight lines $AC, BD$ makes the alternate angles equal.

Therefore from Equal Alternate Angles implies Parallel Lines:

- $AC \parallel BD$

$\blacksquare$

## Historical Note

This proof is Proposition $33$ of Book $\text{I}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 1*(2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions