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

Let the two straight lines *AB* and *BC* meeting one another be parallel to the two straight lines *DE* and *EF* meeting one another not in the same plane.

I say that the plane produced through *AB* and *BC* and the plane produced through *DE* and *EF* do not meet one another.

Draw *BG* from the point *B* perpendicular to the plane through *DE* and *EF* to where it meets the plane at the point *G.*

Draw *GH* through *G* parallel to *ED,* and *GK* parallel to *EF.*

Now, since *BG* is at right angles to the plane through *DE* and *EF,* therefore it makes right angles with all the straight lines which meet it and lie in the plane through *DE* and *EF.*

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

And, since *BA* is parallel to *GH,* therefore the sum of the angles *GBA* and *BGH* equals two right angles.

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* and *BC* which cut one another, therefore *GB* is also at right angles to the plane through *BA* and *BC.*

But planes to which the same straight line is at right angles are parallel, therefore the plane through *AB* and *BC* is parallel to the plane through *DE* and *EF.*

Therefore, *if two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then the planes through them are parallel.*

Q.E.D.