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 Congruence 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