---
book: 11
number: 9
id: "XI.9"
kind: "theorem"
uses: ["[[book-11/proposition-4]]", "[[book-11/proposition-8]]", "[[book-11/proposition-6]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.9"
license: "CC-BY-SA-4.0"
---

# XI.9

*Straight lines which are parallel to the same straight line and are not in the same plane with it are also parallel to one another.*

## Proof

For let each of the straight lines *AB*, *CD* be parallel to *EF*, not being in the same plane with it; I say that *AB* is parallel to *CD*.

For let a point *G* be taken at random on *EF*, and from it let there be drawn *GH*, in the plane through *EF*, *AB*, at right angles to *EF*, and *GK* in the plane through *FE*, *CD* again at right angles to *EF*.

Now, since *EF* is at right angles to each of the straight lines *GH*, *GK*, therefore *EF* is also at right angles to the plane through *GH*, *GK*. [[book-11/proposition-4|XI. 4]]

And *EF* is parallel to *AB*; therefore *AB* is also at right angles to the plane through *HG*, *GK*. [[book-11/proposition-8|XI. 8]]

For the same reason *CD* is also at right angles to the plane through *HG*, *GK*; therefore each of the straight lines *AB*, *CD* is at right angles to the plane through *HG*, *GK*.

But if two straight lines be at right angles to the same plane, the straight lines are parallel; [[book-11/proposition-6|XI. 6]] therefore *AB* is parallel to *CD*.