Line joining Points on Parallel Lines is in Same Plane

Proof

 * Euclid-XI-7.png

Let $AB$ and $CD$ be two straight lines which are parallel.

Consider the plane in which they lie to be the plane of reference.

Let $E$ and $F$ be taken at random on both of them.

Then the straight line $EF$ is to be demonstrated to lie in the same plane as $AB$ and $CD$.

Suppose to the contrary that $EF$ is in a more elevated plane: $EGF$.

Let a plane be drawn through $EGF$.

From, the common section of this plane with the plane of reference is a straight line passing through $E$ and $F$.

Therefore $EF$ and $EGF$ enclose an area, which is impossible.

Therefore the straight line $EF$ joining $E$ and $F$ is in the same plane as $AB$ and $CD$.