Geometry Help: Quadrilaterals

"Quadrilateral" comes from the Latin and means "four-sided." In math (specifically geometry) any four-sided polygon is called a quadrilateral.

But wait a second! What in the world is a "polygon"? Again we must look at our ancient languages, in this case, Greek. In the Greek "poly" means "many" and "gon" comes from the Greek word for "angle", so a polygon is a figure with many angles. The current mathematical definition of a polygon is roughly that 1) it is a flat geometric figure (that is, it is two-dimensional); 2) it is simple (that is, there are no crossed lines); 3) closed (that is, no open spaces), and 4) all the sides are line segments (so no curving lines.)

Your average elementary math student is very familiar with polygons, though he might not call them that. Early on we are taught to recognize the shapes for squares, triangles, stars, and diamonds, all of which are actually polygons. (Notice that crescent moons, circles, and ellipses are NOT polygons, as they have curved or circular sides.) In this article I am going to talk about defining and recognize one subset of polygons, the quadrilaterals.

A quadrilateral, by definition, is a four-sided polygon - ANY four-sided polygon. The math student (particularly the geometry math student) will need to be able to walk through the definition tree of quadrilaterals, as they become more tightly defined.

A quadrilateral will be either what is called a "trapezoid" or a "parallelogram." The term trapezoid comes from the Greek for "table shaped" and is defined as a quadrilateral with EXACTLY ONE pair of parallel sides. This is very important as the geometry student may confuse the fact that a trapezoid has a pair of parallel sides and call it a "parallelogram" by mistake. A parallelogram is defined as a quadrilateral with TWO PAIRS of parallel sides.

A trapezoid is a trapezoid is a trapezoid and usually can't be broken down any further, but we can narrow in defining the parallelograms further either by looking at the length of the four sides or the four interior angles. (Math note: polygons always have the same number of sides as they have interior angles. So all quadrilaterals, because they have four sides, also have four interior angles. All triangles - that is, three sided polygons - also have three interior angles, and so on.)

A parallelogram (that is a quadrilateral with two pairs of parallel sides) that has four right angles is called a "rectangle," a term most math students are familiar with. On the other hand, many math students might not be familiar with the term "rhombus" which describes an equilateral parallelogram (that is a quadrilateral with two pairs of parallel sides and four sides all of equal length).

The last quadrilateral which I would expect all geometry (or any math level) student to recognize is simply called the "square." If you want to impress your math teacher, you could describe a square as a "rhombus with four right angles" or an "equilateral rectangle." A square, therefore, is not only a polygon, but also a quadrilateral, a parallelogram, a rectangle, and a rhombus.

Blessings!

Source
John H. Saxon, Jr. Algebra 1