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- Properties of Polygons
- Grouping Polygons
- Polygon-Classification Game
- More Venn Diagrams

Polygons can be divided into groups according to certain properties.

Concave polygons look like they are collapsed or have one or more angles dented in. Any polygon that has an angle measuring more than 180° is concave. These are concave polygons:

These polygons are not concave:

Regular polygons have sides that are all the same length and angles that are all the same size. These polygons are regular:

The polygons below are not regular. Such polygons are referred to as irregular. **Note 3**

A polygon has line symmetry, or reflection symmetry, if you can fold it in half along a line so the two halves match exactly. The “folding line” is called the line of symmetry.

These polygons have line symmetry. The lines of symmetry are shown as dashed lines. Notice that two of the polygons have more than one line of symmetry.

These polygons do not have line symmetry:

Consider the polygons below:

This diagram shows how these four polygons can be grouped into the categories Concave and Not Concave.

The circles need to overlap. Why? What goes in the middle?

Make a diagram to show how these same four polygons can be grouped into the categories Line Symmetry and Not Concave. Use a circle to represent each category. **Note 4**

Problems B1-B4 adapted from IMPACT Mathematics, developed by Educational Development Center, Inc. pp.50-54. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

You will now play a polygon classification game using Venn diagrams.

Venn diagrams use circles to represent relationships among sets of objects. They are named after John Venn (1834-1923) of England. Venn, a priest and historian, published two books on logic in the 1880s. Venn diagrams can be used to solve certain types of logic puzzles.

As you’re placing polygons in the labeled circles, think which properties each polygon has and then check whether that property is shared with any other of the labels. It might also be helpful to create a grid listing all the polygons and labels and filling it in for easier viewing of the shared properties among polygons.

As a warm-up for the game, put each of the labels Regular, Concave, and Triangle next to one of the circles on the diagram. Place all the polygons in the correct regions of the diagram.

The following activity works best when done in groups. If you are working alone, consider asking a friend or colleague to work with you.

You will need a set of polygons, a set of category labels, and at least three loops of string for the polygon-classification game. Print and cut out the polygons and catagory labels.

Using string, set up a Venn diagram with three circles.

If you are working with a partner or in a small group, choose one member of your group to be the leader, and follow these rules to play the polygon classification game:

- The leader selects three category cards and looks at them without showing them to the other group members.
- The leader uses the cards to label the regions, placing one card face down next to each circle.
- The other group members take turns selecting a polygon, and the leader places the polygon in the correct region of the diagram.
- After a player’s shape has been placed in the diagram, he or she may guess what the labels are. The first player to guess correctly wins.
- Take turns being the leader until each member of the group has had a chance.

If the leader misplaces one or more shapes, your group may have trouble guessing the labels. Encourage everyone to ask the leader to check placement of particular pieces that seem wrong. If your group is stuck, if possible, ask someone from another group to peek at the labels and help the leader move shapes if necessary.

If you are working on your own or to help you get started, look at each of these Venn diagrams and see if you can determine how they should be labeled.

**a.**

**b.**

**c.**

**d.**

**e.**

The following problems further explore the properties of polygons using Venn diagrams:

Use the picture of a Venn diagram below:

**a. **Determine what the labels on this diagram must be.

**b. **Explain why there are no polygons in the overlap of the Label 1 circle and the Label 2 circle.

**c. **Explain why there are no polygons in the Label 3 circle that are not also in one of the other circles.

Create a diagram in which no polygons are placed in an overlapping region (that is, no polygon belongs to more than one category).

Create a diagram in which all of the polygons are placed either in the overlapping regions or outside the circles (that is, no polygon belongs to just one category).

**Note 3**

For each shape, take a moment to discuss or reflect on how it fails to meet the definition of “regular.” For example, in the quadrilateral on the left, all the sides have the same length. But because the angles are not all the same, it is not a regular polygon. In the rectangle, all the angles are the same, but the sides have different lengths. So it is not a regular polygon.

**Note 4**

This is best done as a full-group discussion with an overhead or at the board. Have someone draw the two regions, hold up cutouts of the shapes, and ask the group to decide where each would go. If you are working on an overhead, string is better for creating the regions than drawing because it is easily altered to “fix” the regions to overlap.

See the diagram below. Note that shapes e, f, h, j, and m are neither triangular, regular, nor concave, so they belong outside all three circles of the diagram.

**a.**

**b.**

**c.**

**d.**

**e.**

**a.** Label 1 is Irregular, Label 2 is Regular, and Label 3 is Pentagon.

**b.** The overlap would have to contain polygons that are simultaneously regular and irregular. No such polygons exist!

**c.** Every pentagon is either regular or irregular. For there to be a polygon in the Label 3 circle but not in any others, there would have to be a pentagon which is neither regular nor irregular. This is impossible.

Answers will vary. One possibility is to label the regions Triangle, Quadrilateral, and Pentagon.

Answers will vary. One possibility is Not Triangle, Not Quadrilateral, and Not Hexagon. Even if a shape is one of those three things, it’s sure to not be the other two, so every shape is in the overlap of two circles, and some (pentagons) are in the overlap of all three.