---
book: 11
number: 18
id: "XI.18"
kind: "theorem"
uses: ["[[book-1/proposition-11]]", "[[book-1/proposition-28]]", "[[book-11/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.18"
license: "CC-BY-SA-4.0"
---

# XI.18

*If a straight line be at right angles to any plane*, *all the planes through it will also be at right angles to the same plane.*

## Proof

For let any straight line *AB* be at right angles to the plane of reference; I say that all the planes through *AB* are also at right angles to the plane of reference.

For let the plane *DE* be drawn through *AB*, let *CE* be the common section of the plane *DE* and the plane of reference, let a point *F* be taken at random on *CE*, and from *F* let *FG* be drawn in the plane *DE* at right angles to *CE*. [[book-1/proposition-11|I. 11]]

Now, since *AB* is at right angles to the plane of reference, *AB* is also at right angles to all the straight lines which meet it and are in the plane of reference; [[book-11/definitions#Definition 3|XI. Def. 3]] so that it is also at right angles to *CE*; therefore the angle *ABF* is right.

But the angle *GFB* is also right; therefore *AB* is parallel to *FG*. [[book-1/proposition-28|I. 28]]

But *AB* is at right angles to the plane of reference; therefore *FG* is also at right angles to the plane of reference. [[book-11/proposition-8|XI. 8]]

Now a plane is at right angles to a plane, when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane. [[book-11/definitions#Definition 4|XI. Def. 4]]

And *FG*, drawn in one of the planes *DE* at right angles to *CE*, the common section of the planes, was proved to be at right angles to the plane of reference; therefore the plane *DE* is at right angles to the plane of reference.

Similarly also it can be proved that all the planes through *AB* are at right angles to the plane of reference.

Therefore etc. Q. E. D.