Lines Joining Equal and Parallel Straight Lines

Theorem

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

Proof

Let $AB, CD$ be equal and parallel.

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

Draw $BC$.

From Parallel Implies Equal Alternate Interior Angles, we have $\angle ABC = \angle BCD$.

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 interior angles equal.

Therefore from Equal Alternate Interior Angles Implies Parallel, we have that $AC \| BD$.

$\blacksquare$

Historical Note

This is Proposition 33 of Book I of Euclid's The Elements.