1. The System of Mathematics Good Definitions & Counterexamples
Geometry – Mr. Ferguson

2. Verification of Conditionals
Prior to this point, we have focused on the structure of mathematical statements without an explicit discussion of their truth.
While there are numerous ways in which a conditional can be true, we will focus on one (perhaps two) case this year:
If both the hypothesis and the conclusion of a conditional are true, then the conditional is true.

3. Counterexamples Example:
counterexample = a statement that shows a conjecture is false
In a counterexample, we show that the hypothesis leads to a different conclusion than what was stated in the conditional – this disproves the validity of the argument.
In order to show that a statement is false, we only need one counterexample to exist.
Example:
If the measure of an angle is greater than 60 degrees, then the angle is a right angle.

4. Non-examples vs. Counterexamples
A non-example is an example that is contrary to some definition, theorem, or conjecture.
A counterexample shows that a conditional is false.

5. Sketch an example & a non-example.
right angle
polygon
diameter of a circle

6. Find a counterexample.
If AB = BC, then B is the midpoint of segment AC.
If two angles are complementary, then their non- common rays are perpendicular.
If x is a real number, then x2 is greater than x.

7. good geometric definitions
In geometry, precision of language and vocabulary is a necessity.
Definitions must characterize objects that in a way that excludes non-examples and removes any sense of ambiguity.
Our definitions will be stated as biconditionals, as this will require that the conditional and the converse of a statement are true.

8. What makes a good definition?
Specific and true. Includes enough information to include all examples and exclude all non- examples.
Only uses the undefined terms or terms that we have previously defined.
Stated as a biconditional.
In order for a term to be a good definition, both the conditional and the converse must be true.

9. example: perpendicular lines
Conditional: If two lines are perpendicular, then their intersection forms a right angle.
Converse: If the intersection of two lines forms a right angle, then the lines are perpendicular.
Two lines are perpendicular iff their intersection forms a right angle.

10. Practice with Definitions
Two angles are adjacent iff they have the same vertex.
Write the conditional and the converse from the biconditional above.
Is the biconditional above a good definition of adjacent angles? Why or why not?

11. Practice with Conditionals
If x is odd, then 2x is even.
Is the conditional above true?
Write the converse. Is the converse true?
Write the inverse. Is the inverse true?
Write the contrapositive. Is the contrapositive true?
Can we write a biconditional of the above statement? Why or why not?

12. A right angle has a measure of 90 degrees.
More Practice
A right angle has a measure of 90 degrees.
Write the above statement as a conditional.
Write the converse of the statement.
Write the biconditional associated with the statement.
minus CAP1Plat

13. Biconditional Practice
x = 5 if and only if 2x = 10
Write a conditional and a converse from the biconditional above.

14. Biconditional Practice
The angles of an equilateral triangle are congruent.
Write a biconditional associated with the above statement.
Write a conditional and a converse from the biconditional you wrote.