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**Answers to the HW p. 75 #10-20 even, 25-28 all, 55 & 56**

10. If an object weighs one pound, then it weights 16 ounces. If a fish is a blue trunkfish, then it lives in the waters of a coral reef. True True 18. If 1 is obtuse, then 1 measures 123. If I go to the mall, then it is not raining. 25. One two line a line 55. D D

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**Answers to the HW p. 82 #14 - 26 even, 54 & 55**

14. True true true 20. (conditional) If two angles are congruent, then they have the same measure; (converse) If two angles have the same measure, then they are congruent. 22. (conditional) If two lines are perpendicular, then they intersect to form right angles; (converse) If two lines intersect to form right angles, then the two lines are perpendicular. 24. A 100 angles is obtuse, but doesn’t measure 94. Terry could live in Orlando, FL, not in Tampa, FL. 54. D B

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You may use your notes Identify the conclusion. If the weather is warm, then we should go swimming. Write the contrapositive. If you like purple, then you will like this shirt. Rewrite the biconditional statement as a conditional statement and its converse. Two segments are congruent if and only if they have the same measure. Write the inverse. If you like hockey, then you go to the hockey game. Give a counterexample that demonstrates that the converse of the statement is false. If a vehicle is a car, then it has wheels.

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**Objective: Apply logic to true statements to make valid conclusions**

2.3 Deductive Reasoning Objective: Apply logic to true statements to make valid conclusions

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**Let p be “Today is Tuesday” and q be “There is school.**

1. What is p q ? If today is Tuesday, then there is school. 2. What is ~p ~q ? If today is not Tuesday, then there is no school. 3. What is q p ? If there is school, then today is Tuesday. 4. What is ~q ~p ? If there is no school, then today is not Tuesday.

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**How is this different from Inductive Reasoning?**

Deductive Reasoning Deductive Reasoning uses facts, definitions, and accepted properties to write logical arguments (proofs). How is this different from Inductive Reasoning?

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Law of Detachment If a conditional statement is true and the hypothesis is true, then the conclusion is automatically true.

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**EX 1: Use the Law of Detachment to determine a conclusion.**

p q If a triangle is equilateral, then the measure of each angle is 60. Triangle ABC is an equilateral triangle. p is true The measure of each angle is 60.

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**EX 2: Use the Law of Detachment to determine a conclusion.**

p q If two lines are parallel, then the lines do not intersect. k is parallel to m. p is true k and m do not intersect.

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**EX 3: Use the Law of Detachment to determine a conclusion.**

p If Robby is taller than Ben, then Robby is at least 6 feet tall. Ben is older than Robby. q r is true NO VALID CONCLUSION!

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Law of Syllogism Similar to the transitive property in algebra Has 3 statements: 1st and 2nd are true, 2nd and 3rd are true, so 1st and 3rd must also be true.

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**EX 4: Use the Law of Syllogism to determine a conclusion.**

p q If Donnie asks Pam, then she will say yes. If she says yes, then they will get married. q r If Donnie asks Pam, then they will get married.

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**EX:5 Use the Law of Syllogism to determine a conclusion.**

p q If Tim gets stung by a bee, then he will get very ill. If he gets very ill, then he will go to the hospital. r q If Tim gets stung by a bee, then he will go to the hospital.

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**EX 6: Use the Law of Syllogism to determine a conclusion.**

p q If Jay doesn’t work hard, then he will not play. If he doesn’t play, then he will quit the team. r q If Jay doesn’t work hard, then he will quit the team.

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**EX 7: Use the Law of Syllogism to determine a conclusion.**

p q If Susan screams, then the dog will run away. If the dog licks Susan, then Susan will scream. r p NO VALID CONCLUSION!!

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**Use the true statements to determine whether the conclusion is true or false.**

If it looks like rain, then I will bring my umbrella to school with me. If there are clouds in the sky and the sky is dark, then it looks like rain. If I bring my umbrella to school with me, then I will hang it in the classroom closet. This morning, there are clouds in the sky and the sky is dark. Conclusion: My umbrella is hanging in the classroom closet.

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In order… This morning, there are clouds in the sky and the sky is dark. If there are clouds in the sky and the sky is dark, then it looks like rain. If it looks like rain, then I will bring my umbrella to school with me. If I bring my umbrella to school with me, then I will hang it in the classroom closet. Conclusion: My umbrella is hanging in the classroom closet. TRUE

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**2.4 Reasoning with Algebraic Properties**

Objective: Use Properties of Algebra to explain multi-step equations

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**Algebraic Properties of Equality**

p. 96 and 98 KNOW THEM!

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**Multiplication Property**

Let a, b, and c be real numbers If a = b, then a + c = b + c Addition Property add the same thing to both sides of an eqn. & they will still be equal Subtraction Property If a = b, then a - c = b - c subtract the same thing from both sides of an eqn. & they will still be equal Multiplication Property If a = b, then ac = bc multiply the same thing to both sides of an eqn & they will still be equal Division Property divide the same thing from both sides of an eqn & they will still be equal

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**Substitution Property**

Let a, b, and c be real numbers Reflexive Property For any real number a, a = a A number, length, or measure will always equal itself Symmetric Property If a = b, then b = a If two things are equal then order does not matter Transitive Property If a = b and b = c then a = c This is true by the Law of Syllogism Substitution Property If a = b, then a can be substituted for b in any equation or expression Think about substituting 3 in for x in an equation when x = 3

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**To solve the equation 10y + 5 = 25 ,put these steps in order**

Given y = 2 10y = Subtraction Prop. 10y = 20 10y = 20 Simplify 10y + 5 = 25 Division Property y = 2 Simplify

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**Write a reason for each step.**

Given Distributive Property Addition Property Subtraction Property Subtraction Property Division Property

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**Use the property to complete the statement.**

1. Addition property of equality: If AB = 5, then 10 + AB = ____. 15 2. Multiplication property of equality: If mC = 30, then ____ (mC) = 15 . 1/2 3. Reflexive property of equality: AF = ___. AF 4. Symmetric property of equality: If mDCF = mMJC, then ________________. mMJC = mDCF 5. Transitive property of equality: If YZ = DB and DB = JK, then _________. YZ = JK 6. Substitution property of equality: If MN = 3, then 5(MN) = _____. 5(3)

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**WY = XZ Show that WX = YZ WY = XZ WY = WX + XY XZ = XY + YZ**

Given WY = XZ Segment Addition Postulate WY = WX + XY Segment Addition Postulate XZ = XY + YZ WX + XY = XY + YZ Substitution Prop. Of Equality Subtraction Prop. Of Equality WX = YZ

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**Substitution prop of equality**

Banked turns help the cars travel around the track at high speeds. The angles provide an inward force that helps keep the cars from flying off the track. Given the following information about the four banked turns at the Talladega Superspeedway racetrack in Alabama, find the measure of angle 4 Given Given Substitution prop of equality Subtraction prop of equality = Given Transitive prop of equality Substitution prop of equality

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**p.99–101 (9 problems) #4-8, 10-14 even, 32 a-e**

Homework: p (13 problems) #8-32 even p.99–101 (9 problems) #4-8, even, 32 a-e

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Unit 2: Deductive Reasoning

Unit 2: Deductive Reasoning

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