---
book: 11
number: 14
id: "XI.14"
kind: "theorem"
uses: ["[[book-11/proposition-3]]", "[[book-1/proposition-17]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.14"
license: "CC-BY-SA-4.0"
---

# XI.14

*Planes to which the same straight line is at right angles will be parallel.*

## Proof

For let any straight line *AB* be at right angles to each of the planes *CD*, *EF*; I say that the planes are parallel.

For, if not, they will meet when produced.

Let them meet; they will then make, as common section, a straight line. [[book-11/proposition-3|XI. 3]]

Let them make *GH*; let a point *K* be taken at random on *GH*, and let *AK*, *BK* be joined.

Now, since *AB* is at right angles to the plane *EF*, therefore *AB* is also at right angles to *BK* which is a straight line in the plane *EF* produced; [[book-11/definitions#Definition 3|XI. Def. 3]] therefore the angle *ABK* is right.

For the same reason the angle *BAK* is also right.

Thus, in the triangle *ABK*, the two angles *ABK*, *BAK* are equal to two right angles: which is impossible. [[book-1/proposition-17|I. 17]]

Therefore the planes *CD*, *EF* will not meet when produced; therefore the planes *CD*, *EF* are parallel. [[book-11/definitions#Definition 8|XI. Def. 8]]

Therefore planes to which the same straight line is at right angles are parallel. Q. E. D.