---
book: 11
number: 15
id: "XI.15"
kind: "theorem"
uses: ["[[book-11/proposition-11]]", "[[book-1/proposition-31]]", "[[book-11/proposition-9]]", "[[book-1/proposition-29]]", "[[book-11/proposition-4]]", "[[book-11/proposition-14]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.15"
license: "CC-BY-SA-4.0"
---

# XI.15

*If two straight lines meeting one another be parallel to two straight lines meeting one another*, *not being in the same plane*, *the planes through them are parallel.*

## Proof

For let the two straight lines *AB*, *BC* meeting one another be parallel to the two straight lines *DE*, *EF* meeting one another, not being in the same plane; I say that the planes produced through *AB*, *BC* and *DE*, *EF* will not meet one another.

For let *BG* be drawn from the point *B* perpendicular to the plane through *DE*, *EF* [[book-11/proposition-11|XI. 11]], and let it meet the plane at the point *G*; through *G* let *GH* be drawn parallel to *ED*, and *GK* parallel to *EF*. [[book-1/proposition-31|I. 31]]

Now, since *BG* is at right angles to the plane through *DE*, *EF*, therefore it will also make right angles with all the straight lines which meet it and are in the plane through *DE*, *EF*. [[book-11/definitions#Definition 3|XI. Def. 3]]

But each of the straight lines *GH*, *GK* meets it and is in the plane through *DE*, *EF*; therefore each of the angles *BGH*, *BGK* is right.

And, since *BA* is parallel to *GH*, [[book-11/proposition-9|XI. 9]] therefore the angles *GBA*, *BGH* are equal to two right angles. [[book-1/proposition-29|I. 29]]

But the angle *BGH* is right; therefore the angle *GBA* is also right; therefore *GB* is at right angles to *BA*.

For the same reason *GB* is also at right angles to *BC*.

Since then the straight line *GB* is set up at right angles to the two straight lines *BA*, *BC* which cut one another, therefore *GB* is also at right angles to the plane through *BA*, *BC*. [[book-11/proposition-4|XI. 4]]

But planes to which the same straight line is at right angles are parallel; [[book-11/proposition-14|XI. 14]] therefore the plane through *AB*, *BC* is parallel to the plane through *DE*, *EF*.

Therefore, if two straight lines meeting one another be parallel to two straight lines meeting one another, not in the same plane, the planes through them are parallel. Q. E. D.