If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles.

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 angle *ABC* equals the angle *DEF.*

Cut *BA, BC, ED,* and *EF* off equal to one another, and join *AD, CF, BE, AC,* and *DF.*

Now, since *BA* equals and is parallel to *ED,* therefore *AD* also equals and is parallel to *BE.* For the same reason *CF* also equals and is parallel to *BE.*

Therefore each of the straight lines *AD* and *CF* equals and is parallel to *BE.* But straight lines which are parallel to the same straight line and are not in the same plane with it are parallel to one another, therefore *AD* is parallel and equal to *CF.*

And *AC* and *DF* join them, therefore *AC* also equals and is parallel to *DF.*

Now, since the two sides *AB* and *BC* equal the two sides *DE* and *EF,* and the base *AC* equals the base *DF,* therefore the angle *ABC* equals the angle *DEF.*

Therefore, *if two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles.*

Q.E.D.