Lines Joining Equal and Parallel Straight Lines are Parallel

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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

Euclid-I-33.png

Let $AB, CD$ be equal and parallel.

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


Draw $BC$.

From Parallelism implies Equal Alternate Interior 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 interior angles equal.

Therefore from Equal Alternate Interior Angles implies Parallel Lines:

$AC \parallel BD$

$\blacksquare$


Historical Note

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


Sources