Biconditionals & Counterexamples Geometry Chapter 02 A BowerPoint Presentation.

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Biconditionals & Counterexamples Geometry Chapter 02 A BowerPoint Presentation

Biconditionals We can combine two statements into one biconditional statement when –They are converses of each other, and –They are both true.

Biconditionals If an angle is a right angle, then it has a measure of 90°. If an angle has a measure of 90, then it is a right angle.

Biconditionals If an angle is a right angle, then it has a measure of 90°. If an angle has a measure of 90 °, then it is a right angle. Are these statements converses of each other?

Biconditionals If an angle is a right angle, then it has a measure of 90°. If an angle has a measure of 90, then it is a right angle. Are these statements converses of each other? YES

Biconditionals If an angle is a right angle, then it has a measure of 90°. If an angle has a measure of 90, then it is a right angle. Are these statements both true?

Biconditionals If an angle is a right angle, then it has a measure of 90°. If an angle has a measure of 90, then it is a right angle. Are these statements both true? YES

Biconditionals To make a biconditional, we combine the two statements with the phrase IF AND ONLY IF.

Biconditionals To make a biconditional, we combine the two statements with the phrase IF AND ONLY IF. If an angle is a right angle, then it has a measure of 90°. If an angle has a measure of 90°, then it is a right angle. An angle is a right angle IF AND ONLY IF it has a measure of 90°.

Biconditionals You try to make a biconditional statement from these two statements: If the sum of two angles = 180°, then the angles are supplementary. If two angles are supplementary, then the sum of the angles = 180°. (Remember to use IF AND ONLY IF)

Biconditionals Two angles are supplementary if and only if the sum of the angles = 180°. OR The sum of two angles = 180° if and only if the angles are supplementary.

Biconditionals Definitions must be biconditional. WHY?

Counterexamples People like to give examples to prove why their statements are right. What are some examples your friends, parents, or teachers have used to prove they are right?

Counterexamples A counterexample is an example that is false. You use it to show people they are wrong. If you can give one counterexample to a statement, that statement must be false (it can’t be true or your counterexample would be true!).

Counterexamples What are some counterexamples you’ve used on friends, parents, and teachers to show them their statements were wrong?

Counterexamples What could be a counterexample to the following statement? Everyone loves ice cream.

Counterexamples What could be a counterexample to the following statement? If n is a real number, then

Counterexamples Someone makes a statement and you CAN provide a counterexample. –What does that mean about the original statement?

Counterexamples Someone makes a statement and you CAN provide a counterexample. –What does that mean about the original statement? Someone makes a statement and you CAN’T provide a counterexample. –What does that mean about the original statement?