Biconditionals and Definitions
Recognizing Good Definitions
Investigation: Writing a Definition
- Use the example at the left to identify the figures above that are polyglobs.
- Write a definition of a polyglob by describing what a polyglob is.
In geometry you start with undefined terms such as point, line, and plane whose meanings you understand intuitively. Then you use those terms to define other terms such as collinear points.
A good definition is a statement that can help you identify or classify an object. A good definition has several important components.
- A good definition uses clearly understood terms. The terms should be commonly understood or already defined.
- A good definition is precise. Good definitions avoid words such as large, sort of, and some.
- A good definition is reversible. That means that you can write a good definition as a true biconditional.
Writing a Definition as a Biconditional
Show that this definition of perpendicular lines is reversible. Then write it as a true biconditional.
DefinitionPerpendicular lines are two lines that intersect to form right angles.
ConditionalIf two lines are perpendicular, then they intersect to form right angles.
ConverseIf two lines intersect to form right angles, then they are perpendicular.
The two conditionals—converses of each other—are true, so the definition can be written as a true biconditional.
BiconditionalTwo lines are perpendicular if and only if they intersect to form right angles.
One way to show that a statement is not a good definition is to find a counterexample.
Think about the Venn diagram for "An airplane is a vehicle that flies."
Language Arts Is the given statement a good definition? Explain.
- An airplane is a vehicle that flies. The statement is not a good definition because it is not reversible. A helicopter is a counterexample. A helicopter is a vehicle that flies, but a helicopter is not an airplane.
- A triangle has sharp corners. The statement is not a good definition because it uses the imprecise word sharp, and it is not reversible.
Interactivity Biconditional Statements
Exercises Click here to view the Exercises for this lesson.