Trapezoid

A trapezoid (in British English usually a trapezium) is a quadrilateral with at least one pair of parallel sides. The parallel sides are its bases $a$ and $b$, the other two are its legs, and the distance between the bases is its height $h$. The picture keeps the bases on two parallel lines, so every shape made there is a trapezoid as long as the top base is longer than 0 (at $a = 0$ it becomes a triangle); the readout names the narrower kinds it also belongs to and works out its area.

Yes, with the definition used here. A parallelogram has two pairs of parallel sides, so it certainly has one; so do rectangles, rhombi and squares. Another definition, also in use, asks for exactly one pair, and then a parallelogram is not a trapezoid. This page uses “at least one”: what is proved for trapezoids then holds for parallelograms too, with no separate case. The area formula and the midline $(a + b)/2$ still give the right answers when $a = b$.

An isosceles trapezoid has equal base angles on each base, and so equal legs and equal diagonals. Equal legs alone are not enough: a slanted parallelogram has them but is not isosceles, while a rectangle is. A right trapezoid has a leg perpendicular to the bases; that leg is the height.

The three proofs on the red key rearrange the trapezoid into shapes whose areas are known. Two copies, one turned half a turn about the midpoint of a leg, make a parallelogram with base $a + b$ and height $h$; the trapezoid is half of it. A diagonal cuts it into two triangles of height $h$, one on base $a$ and one on base $b$. Cutting along the midline and turning the top piece gives a parallelogram with base $a + b$ and height $h/2$.

With $a = b$ the shape is a parallelogram, still a trapezoid by this page’s definition, and the formula gives its area $bh$. With $a = 0$ the top base has shrunk to a point: the formula gives $bh/2$, the triangle’s area, but a triangle is not a quadrilateral, so this is where the trapezoids end rather than one more kind of trapezoid. Sliding the top base along its line changes neither base nor the height, so the area stays the same however slanted the shape, even once the height falls outside it.