Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles

Theorem

In the words of Euclid:

If two straight lines meeting one another be parallel to two straight lines meeting one another not in the same plane, they will contain equal angles.

(The Elements: Book $\text{XI}$: Proposition $10$)

Proof

Let $AB$ and $BC$ be straight lines which meet one another.

Let $DE$ and $EF$ be straight lines which meet one another such that $AB$ is parallel to $DE$ and $BC$ is parallel to $EF$.

It is to be demonstrated that $\angle ABC = \angle DEF$.

Let $BA, BC, ED, EF$ be cut off equal to one another.

Let $AC, CF, BE, AD, DF$ be joined.

We have that:

$BA = ED$

and

$BA \parallel ED$

Therefore from Proposition $33$ of Book $\text{I} $: Lines Joining Equal and Parallel Straight Lines are Parallel:

$AD = BE$

and

$AD \parallel BE$

For the same reason:

$CF = BE$

and

$CF \parallel BE$

So each of $AD$ and $CF$ is equal and parallel to $BE$.

But from Proposition $9$ of Book $\text{XI} $: Lines Parallel to Same Line not in Same Plane are Parallel to each other:

$AD = CF$

and

$AD \parallel CF$

We have that $AC$ and $DF$ join $AD$ and $CF$.

Therefore from Proposition $33$ of Book $\text{I} $: Lines Joining Equal and Parallel Straight Lines are Parallel:

$AC = DF$

and

$AC \parallel DF$

So we have that $AB$ and $BC$ are equal and parallel to $DE$ and $EF$.

We also have that $AC = DF$.

So from Proposition $8$ of Book $\text{I} $: Triangle Side-Side-Side Congruence:

$\angle ABC = \angle DEF$

$\blacksquare$

Historical Note

This proof is Proposition $10$ of Book $\text{XI}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions