---
book: 11
number: 6
id: "XI.6"
kind: "theorem"
uses: ["[[book-1/proposition-4]]", "[[book-1/proposition-8]]", "[[book-11/proposition-5]]", "[[book-11/proposition-2]]", "[[book-1/proposition-28]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.6"
license: "CC-BY-SA-4.0"
---

# XI.6

*If two straight lines be at right angles to the same plane*, *the straight lines will be parallel.*

## Proof

For let the two straight lines *AB*, *CD* be at right angles to the plane of reference; I say that *AB* is parallel to *CD*.

For let them meet the plane of reference at the points *B*, *D*, let the straight line *BD* be joined, let *DE* be drawn, in the plane of reference, at right angles to *BD*, let *DE* be made equal to *AB*, and let *BE*, *AE*, *AD* be joined.

Now, since *AB* 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 each of the straight lines *BD*, *BE* is in the plane of reference and meets *AB*; therefore each of the angles *ABD*, *ABE* is right.

For the same reason each of the angles *CDB*, *CDE* is also right.

And, since *AB* is equal to *DE*, and *BD* is common, the two sides *AB*, *BD* are equal to the two sides *ED*, *DB*; and they include right angles; therefore the base *AD* is equal to the base *BE*. [[book-1/proposition-4|I. 4]]

And, since *AB* is equal to *DE*, while *AD* is also equal to *BE*, the two sides *AB*, *BE* are equal to the two sides *ED*, *DA*; and *AE* is their common base; therefore the angle *ABE* is equal to the angle *EDA*. [[book-1/proposition-8|I. 8]]

But the angle *ABE* is right; therefore the angle *EDA* is also right; therefore *ED* is at right angles to *DA*.

But it is also at right angles to each of the straight lines *BD*, *DC*; therefore *ED* is set up at right angles to the three straight lines *BD*, *DA*, *DC* at their point of meeting; therefore the three straight lines *BD*, *DA*, *DC* are in one plane. [[book-11/proposition-5|XI. 5]]

But, in whatever plane *DB*, *DA* are, in that plane is *AB* also, for every triangle is in one plane; [[book-11/proposition-2|XI. 2]] therefore the straight lines *AB*, *BD*, *DC* are in one plane.

And each of the angles *ABD*, *BDC* is right; therefore *AB* is parallel to *CD*. [[book-1/proposition-28|I. 28]]

Therefore etc. Q. E. D.