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 $1$.

$\blacksquare$