---
book: 11
number: 24
id: "XI.24"
kind: "theorem"
uses: ["[[book-11/proposition-16]]", "[[book-11/proposition-10]]", "[[book-1/proposition-34]]", "[[book-1/proposition-4]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.24"
license: "CC-BY-SA-4.0"
---

# XI.24

*If a solid be contained by parallel planes*, *the opposite planes in it are equal and parallelogrammic.*

## Proof

For let the solid *CDHG* be contained by the parallel planes *AC*, *GF*, *AH*, *DF*, *BF*, *AE*; I say that the opposite planes in it are equal and parallelogrammic.

For, since the two parallel planes *BG*, *CE* are cut by the plane *AC*, their common sections are parallel. [[book-11/proposition-16|XI. 16]]

Therefore *AB* is parallel to *DC*.

Again, since the two parallel planes *BF*, *AE* are cut by the plane *AC*, their common sections are parallel. [[book-11/proposition-16|XI. 16]]

Therefore *BC* is parallel to *AD*.

But *AB* was also proved parallel to *DC*; therefore *AC* is a parallelogram.

Similarly we can prove that each of the planes *DF*, *FG*, *GB*, *BF*, *AE* is a parallelogram.

Let *AH*, *DF* be joined.

Then, since *AB* is parallel to *DC*, and *BH* to *CF*, the two straight lines *AB*, *BH* which meet one another are parallel to the two straight lines *DC*, *CF* which meet one another, not in the same plane; therefore they will contain equal angles; [[book-11/proposition-10|XI. 10]] therefore the angle *ABH* is equal to the angle *DCF*.

And, since the two sides *AB*, *BH* are equal to the two sides *DC*, *CF*, [[book-1/proposition-34|I. 34]] and the angle *ABH* is equal to the angle *DCF*, therefore the base *AH* is equal to the base *DF*, and the triangle *ABH* is equal to the triangle *DCF*. [[book-1/proposition-4|I. 4]]

And the parallelogram *BG* is double of the triangle *ABH*, and the parallelogram *CE* double of the triangle *DCF*; [[book-1/proposition-34|I. 34]] therefore the parallelogram *BG* is equal to the parallelogram *CE*.

Similarly we can prove that *AC* is also equal to *GF*, and *AE* to *BF*.

Therefore etc. Q. E. D.