LESSON 14 Disproving Conjectures With Counterexamples.

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This lesson defines what a proof is and shows how to write a proof for a given hypothesis and conclusions.

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Presentation transcript:

LESSON 14 Disproving Conjectures With Counterexamples

Review of previous vocabulary Postulate Theorem Conjecture Hypothesis Conclusion Counterexample  a statement that is know to be true without proof  a statement that is proven to be true  a statement that is assumed to be true  follows “if” of a conditional statement  follows “then” of a conditional statement  an example that proves a conjecture or statement is false

More on counterexamples Counterexamples must show the conclusion is false when the hypothesis is true If a fruit is an apple, then it is green. Hypothesis → a fruit is an apple Conclusion → it is green Why is a watermelon not a good counterexample? You must have true hypothesis and a false conclusion While a watermelon is green, it is not an apple What would be a good counterexample? red apple

If two lines are both intersected by a transversal, then they are parallel. Identify hypothesis, conclusion and find a counterexample Hypothesis: two lines are intersected by a transversal Conclusion: they are parallel

If a triangle is not obtuse, then it is acute. Identify hypothesis, conclusion and find a counterexample Hypothesis: a triangle is not obtuse Conclusion: it is acute

Some statements are not written as a conditional statement

Some conditional statements just need to be solved to disprove

Use the data in the table to prove the conjecture is false If the temperature is hotter, then the average amount of rainfall is less. 82º is hotter than 76º but the rainfall is greater than it is at 76º.

Find a counterexample or explain why these statements are false If a person live in the US, then he or she experiences four distinct seasons. Hawaii The number of days in a year is 365. Leap year has 366 days If students visit the zoo, then they will see a polar bear. Not all zoos have polar bears All vehicles have 4 wheels. Motorboat

Questions/Review Just like before answering questions with just True or False does not take much effort So be sure to take care in your explanations or counterexamples because that is also graded on your assignments and tests A good explanation with a wrong answer will commonly earn more points than a right answer with a poor or no explanation