# Quadrilaterals Exploration

**Objective:**

- Familiarize yourself with the different classes of quadrilaterals.
- Think critically about what exactly a definition tells us.
- Create good, informative examples as well as counter-examples

Polygons, like living things, can be organized by a hierarchy of classes. Some members of the quadrilateral class are subclasses of other types of quadrilaterals. Squares, for example, are a subclass of rectangles, because a square is a rectangle whose sides are all the same length. Rectangles, however, are not a subclass of squares, because not all rectangles are squares.

One quadrilateral may be a subclass of more than one other kind of quadrilateral. Squares are a subclass of both rhombuses and rectangles. But since most rhombuses are not rectangles, and most rectangles are not rhombuses, neither is a subclass of the other. Below, read about a dozen types of quadrilaterals by definition, all have four sides, but each has unique qualities. Then answer the questions below.

- Cv: Convex quadrilateral whose angles are all less than 180 degrees.
- Cy: Cyclic quadrilateral: a quadrilateral whose corners all lie on a single circle.
- K: Kite: a quadrilateral with two pairs of adjoining sides of equal length.
- P: Parallelogram: a quadrilateral whose opposite sides are parallel.
- Q: Quadrilateral: a polygon with four sides.
- Rc: Rectangle: a quadrilateral with four 90° angles.
- Rh: Rhombus: a quadrilateral whose sides are all the same length.
- S: Square: a quadrilateral with four sides of equal length and four 90° angles.
- T: Trapezoid: a quadrilateral having exactly two parallel sides.
- U1: Unnamed quadrilateral 1: has opposite angles of equal measure.
- U2: Unnamed quadrilateral 2: has at least one pair of opposite angles both 90°.
- U3: Unnamed quadrilateral 3: has at least one pair of opposite sides of equal length.

### Questions

- Draw representative illustrations (several) of these types of quadrilaterals.
- Which two definitions describe exactly the same type of quadrilateral?
- Besides Q, which two types of quadrilaterals are not necessarily convex?
- Quadrilaterals of class U2 (one pair of opposite angles both 90°) are a subclass of which two other types of quadrilaterals, not counting Q?
- Can you find a chain of five of the quadrilaterals in which each type is a subclass of the next? Hint: The first is S and the last is Q.
- Can you find one set of four quadrilaterals in which no one quadrilateral is a subclass of another? Hint: Try drawing pictures.

BONUS QUESTION: Can you find--or define--a fifth type of quadrilateral that is neither a subclass nor a super-class of these four quadrilaterals?

**Handin:**
A sheet with answers to all questions.

Some portions originally from ^{[1]} and are Copyright 2000 The Walt Disney Company, used with permission.

- ↑ Scott Kim, Discover "Bogglers", July 2000