---
book: 11
number: 13
id: "XI.13"
kind: "theorem"
uses: ["[[book-11/proposition-3]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.13"
license: "CC-BY-SA-4.0"
---

# XI.13

*From the same point two straight lines cannot be set up at right angles to the same plane on the same side.*

## Proof

For, if possible, from the same point *A* let the two straight lines *AB*, *AC* be set up at right angles to the plane of reference and on the same side, and let a plane be drawn through *BA*, *AC*; it will then make, as section through *A* in the plane of reference, a straight line. [[book-11/proposition-3|XI. 3]]

Let it make *DAE*; therefore the straight lines *AB*, *AC*, *DAE* are in one plane.

And, since *CA* is at right angles to the plane of reference, it will also make right angles with all the straight lines which meet it and are in the plane of reference. [[book-11/definitions#Definition 3|XI. Def. 3]]

But *DAE* meets it and is in the plane of reference; therefore the angle *CAE* is right.

For the same reason the angle *BAE* is also right; therefore the angle *CAE* is equal to the angle *BAE*.

And they are in one plane: which is impossible.

Therefore etc. Q. E. D.