Distance between Two Parallel Straight Lines is Everywhere the Same
Let perpendiculars $EF$ and $GH$ be drawn from $AB$ to $CD$, where $E, G$ are on $AB$ and $F, H$ are on $CD$.
Then $EF = GH$.
That is, the distance between $AB$ and $CD$ is the same everywhere along their length.
$\angle EFH, \angle FEG, \angle EGH, \angle FHG$ are all right angles.
Then $EF$ and $GH$ are parallel.
Thus $\Box EFHG$ is by definition a parallelogram.
From Opposite Sides and Angles of Parallelogram are Equal it follows that $EF = GH$.