Question

Why is a trapezoid a quadrilateral, but a quadrilateral is not always a trapezoid?

Answer 1

When you consider the relationship between two shapes, it is useful to do so from both standpoints, i.e. necessary vs. sufficient.

Necessary - `A` cannot exist without the qualities of `B`.
Sufficient - The qualities of `B` sufficiently describe `A`.

`A` = trapezoid
`B` = quadrilateral

Questions you might want to ask:

1. Can a trapezoid exist without possessing the qualities of a quadrilateral?
2. Are the qualities of a quadrilateral sufficient to describe a trapezoid?

Well, from these questions we have:

1. No. A trapezoid is defined as a quadrilateral with two parallel sides. Therefore, the quality of "quadrilateral" is necessary, and this condition is satisfied.
2. No. Any other shape can have four sides, but if it does not have (at least) two parallel sides, it cannot be a trapezoid. An easy counterexample is a boomerang, which has exactly four sides, but none of them are parallel. Therefore, the qualities of a quadrilateral do not sufficiently describe a trapezoid and this condition is not satisfied.

Some crazy examples of quadrilaterals:

This means that a trapezoid is too specific of a quadrilateral that merely having the quality of "quadrilateral" does not guarantee the quality of "trapezoid".

Overall, a trapezoid is a quadilateral, but a quadrilateral doesn't have to be a trapezoid.