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2.5 - I F -T HEN S TATEMENTS AND D EDUCTIVE R EASONING Homework #6

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M ATHEMATICAL S TATEMENTS A Conditional Statement (if-then statement) has two parts. The “if” part contains the hypothesis. The “then” part contains the conclusion.

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Example 1 Identify the Hypothesis and Conclusion Identify the hypothesis and the conclusion of the if-then statement. If I pass the driving test, then I will get my driver’s license.

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Example 2 Write If-Then Statements Rewrite the statement as an if-then statement. a. Every game on my computer is fun to play. I will buy the CD if it costs less than $15. b. SOLUTION a. If a game is on my computer, then it is fun to play. If the CD costs less than $15, then I will buy it. b.

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D EDUCTIVE R EASONING Deductive Reasoning uses facts, definitions, accepted properties, and the laws of logic to make a logical argument. This form of reasoning differs from inductive reasoning, in which previous examples and patterns are used to form a conjecture.

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L AWS OF L OGIC

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If the car’s battery is dead, then the car won’t start. The car’s battery is dead. The car won’t start. If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of. When it snows, the temperature is less than 32 degrees. It is 20 degrees outside. NO CONCLUSION! If you do not get enough sleep, you will be tired. Snoopy is tired. NO CONCLUSION! Law of Detachment

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L AW OF D ETACHMENT Law of Detachment: If p implies q, and p is true….. q is true! If it is rainy, then I stay inside. It is raining. Conclusion: I stay inside.

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In order for Law of Detachment to apply, the second statement MUST be the hypothesis of the first! If I eat, then I am hungry. I’m hungry. NO CONCLUSION!

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Example 3 Use the Law of Detachment What can you conclude from the following true statements? If you wash the cotton T-shirt in hot water, then it will shrink. You wash the cotton T-shirt in hot water. SOLUTION The hypothesis (you wash the cotton T-shirt in hot water) of a true if- then statement is true. By the Law of Detachment, the conclusion must be true. ANSWER You can conclude that the cotton T-shirt shrinks.

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Checkpoint If-Then Statements and the Law of Detachment Rewrite the statement as an conditional statement. Then underline the hypothesis and circle the conclusion. All teachers at East High School have taught for at least 5 years. 1. An angle is obtuse if its measure is 170°. 2. ANSWER If a teacher works at East High School, then he or she has taught for at least 5 years. ANSWER If the measure of an angle is 170°, then the angle is obtuse.

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Checkpoint If-Then Statements and the Law of Detachment What can you conclude from the given true statements? If x has a value of 7, then 2 x – 3 has a value of 11. The value of x is 7. 3. If you study at least two hours for the test, then you will pass the test. You study three hours for the test. 4. ANSWER 2 x – 3 has a value of 11. ANSWER You will pass the test.

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Example 4 Use the Law of Detachment Which argument is correct? Explain your reasoning. If two angles are vertical angles, then they are congruent. 1 and 2 are congruent. So, 1 and 2 are vertical angles. Argument 1: If two angles are vertical angles, then they are congruent. 1 and 2 are vertical angles. So, 1 and 2 are congruent. Argument 2: SOLUTION Argument 2 is correct. The hypothesis (two angles are vertical angles) is true, which implies that the conclusion (they are congruent) is true. You can use the following counterexample to show that Argument 1 is false. In the diagram at the right, 1 2, but they are not vertical angles.

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Checkpoint Use the Law of Detachment If two angles from a linear pair, then they are supplementary. 3 and 4 from a linear pair. So, 3 and 4 are supplementary. Argument 1 : If two angles from a linear pair, then they are supplementary. 3 and 4 are supplementary. So, 3 and 4 from a linear pair. Argument 2 : Which argument is correct? Explain your reasoning.5. ANSWER Argument 1 is correct. In Argument 1, the hypothesis is true, which implies that the conclusion is true. The following counterexample shows that Argument 2 is false: Two nonadjacent angles with measures 40° and 140° are supplementary, but they do not form a linear pair.

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Example 5 Use the Law of Syllogism Write the statement that follows from the pair of true statements. If the daily high temperature is 32°F or less, then the water in the pipe is frozen. If the water in the pipe is frozen, then the pipe will break. SOLUTION Use the Law of Syllogism. If the daily high temperature is 32°F or less, then the pipe will break.

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If a person eats chicken, then they eat meat. If a person eats meat, then they are not a vegetarian. Derek eats chicken. Derek is not a vegetarian. On Sundays, GHS is not in session. When GHS is not in session, no homework is assigned. Yesterday was Sunday. There was no homework assigned yesterday. If a number is prime, then it does not have repeated factors. If a number does not have repeated factors, then it is not a perfect square. 2161 is a prime number. 2161 is not a perfect square. Law of Syllogism

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Checkpoint Use the Law of Syllogism Write the statement that follows from the pair of true statements. If the ball is thrown at the window, it will hit the window. If the ball hits the window, then the window will break. 6. ANSWER If the ball is thrown at the window, then the window will break.

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