2.14: Classifying Quadrilaterals
Learning Objectives
 Use a Venn diagram to classify quadrilaterals.
Using a Venn Diagram for Classification
You have just explored many different rules and classifications for quadrilaterals. There are different ways to collect and understand this information, but one of the best methods is to use a Venn Diagram. Venn Diagrams are a way to classify objects according to their properties.
A ______________ Diagram is a visual way to classify objects according to their properties.
Think of a snake, which is one type of reptile. All snakes are reptiles, but there are also some reptiles that are not snakes (like turtles and lizards).
Now let’s apply the use of a Venn Diagram to Geometry. Think of a rectangle. A rectangle is a type of parallelogram, but not all parallelograms are rectangles. Here’s a simple Venn Diagram of that relationship:
Notice that all rectangles are parallelograms, but not all parallelograms are rectangles.
If an item falls into more than one category, it is placed in the overlapping section between the appropriate classifications. For example, it is possible for an animal to both have legs and be a reptile. An example of an animal that both has legs and is a reptile is a turtle.
There are some reptiles that do not have legs, like snakes.
There are also some animals with legs that are not reptiles, like a cat.
Finally, there are also some animals that neither have legs, nor are reptiles. These animals would be placed outside of the circles. An example of an animal that is neither a reptile nor has legs is a jellyfish.
To begin organizing the information for a Venn diagram, you can analyze the quadrilaterals we have discussed thus far by three characteristics: parallel sides, congruent sides, and congruent angles.
Below is a table that shows how each quadrilateral fits these characteristics:
Shape  Number of pairs of parallel sides  Number of pairs of congruent sides  Four congruent angles 

Parallelogram  2  2  No 
Rhombus  2  2  No 
Rectangle  2  2  Yes 
Square  2  2  Yes 
Kite  0  2  No 
Trapezoid  1  0  No 
Isosceles trapezoid  1  0  No 
We can choose any characteristics we want to make our Venn Diagram. Let’s make a Venn Diagram based on the number of pairs of parallel sides a quadrilateral has. There will be three main categories:
 Has two pairs of parallel sides (parallelogram, rhombus, rectangle, square)
 Has one pair of parallel sides (trapezoid, isosceles trapezoid)
 Has no pairs of parallel sides (kite)
Example 1
Organize the classification information in the table above in a Venn Diagram.
To begin a Venn Diagram, you must first draw a large ellipse representing the biggest category. In this case, that will be quadrilaterals.
Now, we can add in classes of quadrilaterals.
 The first class of quadrilaterals is the one with two pairs of parallel sides: parallelograms.
 The second class will be quadrilaterals with one pair of parallel sides: trapezoids.
 Finally, the third class will be quadrilaterals with no parallel sides: kites.
Zoom in on the parallelogram oval. There are several types of parallelograms:

Squares, rectangles, and rhombi are all types of parallelograms.
 Some rhombuses are rectangles. These rhombuses are called squares.
 Also, under the category of trapezoids we need to add isosceles trapezoids.
The completed Venn diagram is like this:
You can use this Venn Diagram to quickly answer questions. For instance, is every square a rectangle? (Yes.) Is every rhombus a square? (No, but some are.)
Reading Check:
Use the Venn Diagram above to answer the following questions.
1. True or False: Rhombuses are parallelograms.
2. True or False: Kites are trapezoids.
3. True or False: Some rectangles are also rhombuses.
4. True or False: Some trapezoids are not isosceles.
Graphic Organizer for Lesson 11
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