# What is a Trapezoid? (Part 1)

Alignments to Content Standards: 4.G.A.2

1. Say what a trapezoid is in your own words. Compare your definition with a partner.
2. Is this parallelogram a trapezoid according to your definition? Explain.

## IM Commentary

The purpose of this task is for students to articulate a definition for a trapezoid. There are two competing definitions for "trapezoid":

• The exclusive definition of a trapezoid states that a trapezoid has exactly one pair of opposite sides parallel.

• The inclusive definition states that a trapezoid has at least one pair of opposite sides parallel.

Sometimes people say trapezoids "have one pair of opposite sides parallel," which leaves it ambiguous whether there can be more than one or not. The second part of the task pushes students to be clear about which version they intend. Because of the care students need to take with definitions, this task draws heavily on MP6, Attend to precision.

After students have articulated definitions for themselves or with a partner, the class should discuss the definition together. The class should decide on a single definition that they all agree on, as the point of having clearly articulated definitions is that we all know we are talking about the same thing. While both definitions are legitimate, the benefit to the inclusive definition is that any theorem proved true for a trapezoid is also true for a parallelogram. Furthermore, in their study The Classification of Quadrilaterals (Information Age Publishing, 2008), Usiskin et al. conclude,

The preponderance of advantages to the inclusive definition of trapezoid has caused all the articles we could find on the subject, and most college-bound geometry books, to favor the inclusive definition.

The inclusive definition sets up a relationship between parallelograms and trapezoids that is exactly analogous to to the relationship between squares and rectangles; the definition for rectangles includes squares in the same way that the inclusive definition of trapezoids includes parallelograms.