The purpose of this activity is for students to define a trapezoid and to explore two definitions for a trapezoid. The exclusive definition of a trapezoid states that a trapezoid has exactly one pair of opposite sides that are parallel. The inclusive definition states that a trapezoid has at least one pair of opposite sides that are parallel. Students choose whichever one makes sense to them and look at a parallelogram through the lens of that definition. In the Activity Synthesis, students see how the different definitions impact how a trapezoid fits into the hierarchy of quadrilaterals (MP7). This activity also gives students a chance to plot and label coordinates on the coordinate grid.

• Display: isosceles trapezoid and non-isosceles trapezoid from student solution or use student work
• “How are these two shapes the same? How are they different?” (They both have a pair of parallel sides. One has a pair of equal sides. The other one does not.)
• Display parallelogram from the last problem in student book and this text:

  “A trapezoid . . .”

  “. . . has exactly one pair of opposite sides parallel.”

  “. . . has at least one pair of opposite sides parallel.”

• “According to which definition is this shape a trapezoid? Why?” (the second definition because it has 2 pairs of parallel sides)
• “We’ll continue to explore these two definitions further in the next activity.”

The purpose of this activity is to further explore the two definitions of trapezoids and the hierarchy of quadrilaterals. Students evaluate different statements relating trapezoids and parallelograms deciding whether they are true or false with each definition. The Activity Synthesis establishes the convention for these materials that a trapezoid is a quadrilateral with at least one pair of parallel sides. As students discuss and justify their decisions, they reason clearly using the 2 definitions of trapezoid (MP6).

If students only sort some of the statements correctly, consider asking:

• “How did you decide to sort the statements?”
• Show one statement at a time. “Is this statement true of all the shapes in one of these groups?”

• Invite previously selected students to share.
• “Some people use the first definition of the trapezoid. We will be using the second definition.”
• Display Venn diagrams in the student book.
• “What does each diagram mean?” (Definition 2 shows that a parallelogram is a trapezoid but a trapezoid doesn’t have to be a parallelogram. Definition 1 shows that trapezoids and parallelograms are distinct: a parallelogram can't be a trapezoid and a trapezoid can't be a parallelogram.)
• “Which diagram matches the definition of trapezoid we will use?” (The one on the right, Definition 2, because if a shape is a parallelogram it is also a trapezoid, but if a shape is a trapezoid, it doesn’t have to be a parallelogram.)
• Consider asking students to draw:
  • A trapezoid that is also a parallelogram.
  • A trapezoid that is not a parallelogram.