# Common Sections of Parallel Planes with other Plane are Parallel

## Theorem

In the words of Euclid:

*If two parallel planes be cut by any plane, their common sections are parallel.*

(*The Elements*: Book $\text{XI}$: Proposition $16$)

## Proof

Let $AB$ and $CD$ be two planes which are parallel.

Let $AB$ and $CD$ be cut by the plane $EFGH$.

Let:

- $EF$ be the common section of $EFGH$ and $AB$

and:

- $GH$ be the common section of $EFGH$ and $CD$.

It is to be shown that $EF$ is parallel to $GH$.

Suppose that $EF$ and $GH$ are not parallel.

Then when produced, they will meet, either in the direction of $F$ and $H$, or in the direction of $E$ and $G$.

Without loss of generality, suppose they meet at $K$ when produced in the direction of $F$ and $H$.

We have that $EFK$ is in the plane $AB$.

Therefore from Proposition $1$ of Book $\text{XI} $: Straight Line cannot be in Two Planes:

But $K$ is one of the points of $EFK$.

Therefore $K$ is in the plane $AB$.

For the same reason, $K$ is also in the plane $CD$.

Therefore the planes $AB$ and $CD$ will meet when produced.

But they do not meet, because they are parallel.

Therefore $EF$ and $GH$ will not meet when produced in the direction of $F$ and $H$.

By the same argument, $EF$ and $GH$ will not meet when produced in the direction of $E$ and $G$.

But from Book $\text{I}$ Definition $23$: Parallel Lines:

- $EF$ is parallel to $GH$.

$\blacksquare$

## Historical Note

This proof is Proposition $16$ 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