Equivalence of Definitions of Trapezium
Theorem
The following definitions of the concept of Trapezium are equivalent:
Definition $1$
A trapezium is a quadrilateral which has exactly one pair of sides that are parallel.
Definition $2$
A trapezium is a quadrilateral which has $2$ parallel sides whose lengths are unequal.
Proof
Definition $(1)$ implies Definition $(2)$
Let $T$ be a trapezium by definition $1$.
Then by definition $T$ has exactly one pair of sides that are parallel.
Aiming for a contradiction, suppose those parallel sides of $T$ were the same length.
Then by Quadrilateral is Parallelogram iff One Pair of Opposite Sides is Equal and Parallel, $T$ is a parallelogram.
But $T$ is a trapezium, and so not a parallelogram.
Hence the two parallel sides of $T$ are unequal.
That is, $T$ is a trapezium by definition $2$.
$\Box$
Definition $(2)$ implies Definition $(1)$
Let $T$ be a trapezium by definition $2$.
Then by definition $T$ has a pair of parallel sides, $AB$ and $CD$ say, that are unequal.
Aiming for a contradiction, suppose the other sides of $T$ were parallel.
Then by definition $T$ would be a parallelogram.
Hence by Opposite Sides and Angles of Parallelogram are Equal, $AB = CD$.
This contradicts our supposition that $AB$ and $CD$ are unequal.
Hence by Proof by Contradiction $T$ can have only one pair of parallel sides.
Hence $T$ is a trapezium by definition $1$.
$\blacksquare$