Get down with the lingo
ConcaveDescribes a quadrilateral or polygon that has a cavity, or a cave. It means that at least one of the polygon's diagonals lies partly or entirely outside of the figure.
ConvexDescribes a quadrilateral or polygon that doesn't have a cavity. It means that the polygon has all diagonals completely contained within the figure.
Cross DiagonalThe smaller of the two diagonals of a kite, and it doesn't necessarily involve any symmetry. It is, however, bisected by the main diagonal. So there's that.
Cyclic QuadrilateralA quadrilateral whose vertices all lie on a circle. It has the peculiar property of having supplementary opposite angles.
DiagonalA segment that connects the 2 pairs of opposite vertices in a quadrilateral. In polygons with more sides, a diagonal connects any two vertices that are not right next to each other.
Isosceles TrapezoidA trapezoid whose two non-parallel sides (legs) are congruent. Its two pairs of base angles are also congruent too, much like those of an isosceles triangle.
KiteA quadrilateral with two distinct sets of congruent consecutive sides. (They share an angle.) It has perpendicular diagonals and is perfect for windy days.
Main DiagonalThe larger of the two diagonals of a kite. It's the diagonal about which there is a line of symmetry.
MedianA segment parallel to the bases of a trapezoid that connects the midpoints of the non-parallel sides. A trapezoids median also has a length that's the average of the two bases.
ParallelogramA quadrilateral in which both pairs of opposite sides are parallel. Consequently, they're also congruent. And their opposite angles are congruent. And their consecutive angles are supplementary. And their diagonals bisect each other. And they like long romantic walks on the beach and reading Danielle Steel novels.
QuadrilateralA polygon with four sides. We can remember this easily because "quad" means "four" and "lateral" means "side."
RectangleA parallelogram with four right angles. Also known as "The Equiangulizer" because any equiangular quadrilateral is automatically a rectangle.
RhombusA quadrilateral whose four sides are all equilateral. Rhombi (that's the plural of rhombus) have all the properties of parallelograms, too.
SquareA rectangle with four congruent sides. It's also got that 90° angles thing going on, plus the bisected congruent perpendicular diagonals. It's also the quadrilateral worthy of the title "regular."
TrapezoidA quadrilateral with only one set of parallel sides (called the "bases"). Trapezoids absolutely cannot have two sets of parallel sides. That's a big fat no-no.