# Proposition 10

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.

I.3

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

I.33

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.

XI.9

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.

I.33

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

I.8

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.

## Guide

Of course it is necessary to be careful about in which directions the lines head. If one is changed to head into the opposite direction, then the angles won’t be equal but supplementary instead.

#### Use of this proposition

This proposition is used in the proofs of propositions XI.24 and XII.3.