---
book: 11
number: 7
id: "XI.7"
kind: "theorem"
uses: ["[[book-11/proposition-3]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.7"
license: "CC-BY-SA-4.0"
---

# XI.7

*If two straight lines be parallel and points be taken at random on each of them*, *the straight line joining the points is in the same plane with the parallel straight lines.*

## Proof

Let *AB*, *CD* be two parallel straight lines, and let points *E*, *F* be taken at random on them respectively; I say that the straight line joining the points *E*, *F* is in the same plane with the parallel straight lines.

For suppose it is not, but, if possible, let it be in a more elevated plane as *EGF*, and let a plane be drawn through *EGF*; it will then make, as section in the plane of reference, a straight line. [[book-11/proposition-3|XI. 3]]

Let it make it, as *EF*; therefore the two straight lines *EGF*, *EF* will enclose an area: which is impossible.

Therefore the straight line joined from *E* to *F* is not in a plane more elevated; therefore the straight line joined from *E* to *F* is in the plane through the parallel straight lines *AB*, *CD*.

Therefore etc. Q. E. D.