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1 Geometry UNIT 2 Reasoning and Proof Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 11 Inductive Reasoning and Conjecture Page 93-95: odds, 31-34, 36, 38, 47a Logic Page : 17, 19, 21, 31, 33 Conditional/Biconditional/Inverse/Converse/Contrapositive Page 109-: odd, odd, 47, 48 Review for QUIZ QUIZ Postulates and Paragraph Proofs Page 129: 16, 17, 20, 24, 28, 29, 30*(typo in book) Algebraic Proof Page : 9-11, 13-17, 25, 26 Proving Segment Relationships Page : 9 12, 15 (Do most in class) Proving Angle Relationships Worksheet Review Worksheet and Page : #35, 37, 38 TEST

2 VOCABULARY Inductive Reasoning Conjecture Counterexample Statement Truth Value Compound Statement Negation Conjunction Disjunction Truth Table Conditional Hypothesis Conclusion Converse Inverse Contrapositive Logically Equivalent Deductive Reasoning Law of Detachment Postulate Axiom Two-Column (Formal) Proof Properties of Real Numbers Addition Property Subtraction Property Multiplication Property Division Property Reflexive Property Symmetric Property Transitive Property Substitution Property Distributive Property Postulates/Axioms (accepted without proof) Segment Addition Angle Addition Theorems (can be proven) Supplement Theorem Complement Theorem Vertical Angles Theorem Right Angle Thereoms

3 Inductive Reasoning and Conjecture Inductive Reasoning: Conjecture: Step 1: Look for a pattern Step 2: Make a conjecture Ex 1: Write a conjecture that describes the pattern in the sequence. Then use your conjecture to fine the next item in the sequence. a. Movie show times: 8:30 a.m., 9:45 a.m., 11:00a.m., 12:15 p.m., b. 10, 4, -2, -8, Ex 2: Make a conjecture about each value or geometric relationship. List or draw some examples to support your conjecture. a. The sum of two odd numbers b. Segments joining opposite vertices of a rectangle Counterexample: Ex 4: Find a counterexample to show that the conjecture is false. a. If n is a real number, then n 2 > n. b. Jf JK = KL, the K is the midpoint of JL. c. If n is a real number, then -n is a negative.

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5 Sentences, Statements, and Truth Values, Connectives and Negations Mathematical Sentence: A sentence that states a fact or contains a complete idea. Truth Value: True or False. Statement: A mathematical sentence that is either true or false. Open Sentence: A mathematical sentence that contains a variable. Closed Sentence (Statement): A sentence that can be judged as either true or false. Compound Statement: Two or more statements joined by and or or. Ex 1: (Open) Liver tastes really good. Truth value? Ex 2: (Open) 3x + 2 = 17 Variable? Ex 3: (Open) He is my friend. Variable? Ex 4: (Closed) 3(3) + 2 = 17 Truth Value? Ex 5: (Closed) 3(5) + 2 = 17 Truth Value? Negation: Symbol ~ p: There are seven days in a week. ~p: Conjunction: Symbol Λ Disjunction: Symbol V p q p Λ q T T T F F T F F p q p V q T T T F F T F F p: A right angle measures 90 o. q: Supplementary angles have a sum of 180 o. pλq : Truth Value: ~p V q: Truth Value:

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7 Conditional/Biconditional/Inverse/Converse/Contrapositive Conditional: Symbol (Can also stand for implies ) Hypothesis: Conclusion: p: You pass the test. q: I buy you a pizza. p q: p ~q: p q p q Related Conditionals: a. Inverse: b. Converse: c. Contrapositive d. Logically Equivalent Statements: Biconditional: Symbol p: There is no school. q: It is Saturday. p q: p ~q: p q p q

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9 Reasoning and Proof QUIZ REVIEW Name Use the information below to: a. Write a compound statement in symbolic form. b. Tell whether the compound sentence is true or false. c. For # 3 4, write the inverse, converse, and contrapositive in symbolic form. Let m represent: Let s represent: Let b represent: Let a represent: A segment is bisected at its midpoint. (True) Congruent segments are equal in length. (True) An angle bisector forms two congruent angles. (True) Congruent angles are equal in measure. (True) 1. A segment is bisected at its midpoint or congruent segments are equal in length. Symbolic Form: Truth Value: 2. If an angle bisector does not form two congruent angles then congruent angles are equal in measure. Symbolic Form: Truth Value: Inverse: Converse: Contrapositive: Truth Value: Truth Value: Truth Value: 3. If you study, then you pass the test. (Hyp. is true, conc. is true). Hypothesis: Conclusion: Inverse: Truth Value: Converse: Truth Value: Contrapositive: Truth Value:

10 Postulates and Paragraph Proofs Words Through any two points, there is exactly one line. Picture < > A B C D Through any three noncollinear points there is exactly one plane. A C B m A line contains at least two points. < > A B C D A plane contains at least 3 noncollinear points. n A C B If two point lie in a plane, then the entire line containing those points lies in that plane. A B If two lines intersect, then their intersection is exactly one point. m n X If two planes intersect, then their intersection is a line. Proof: Theorem: Midpoint Theorem: If M is the midpoint of AB, then AM = MB

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12 ALGEBRAIC PROOFS Property of Real Numbers Example Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality Distributive Property of Equality Example: Solve -5(x + 4) = 70 with justification at each step. -5(x + 4) = 70-5x + (-5 4) = 70-5x 20 = 70-5x 20 = x = 90-5x = x = -18

13 8. If -4(x 3) + 5x = 24, then x = 12.

14 Geometry Unit 2 Day 7 Proving Segment Relationships Property Example Reflexive Property of Congruence Symmetric Property of Congruence Transitive Property of Congruence Substitution Property Segment Addition Postulate A whole is equal to the sum of all its parts. Segment Subtraction Postulate

15 1. Line segment AE and DB intersect at C. (DRAW PICTURE) Given: C is the midpoint of AE. C is the midpoint of BD. AE = BD Prove: AC = CD Statement Reason 2. In ΔJNL, point M is on JN and point K is on JL. (DRAW PICTURE) Given: LK = NM, KJ = MJ Prove: LJ = NJ Statement Reason

16 Geometry Unit 2 Day 8 Proving Angle Relationships Complementary Angles Complement Theorem Congruent Complements Theorem Supplementary Angles Supplements Theorem Congruent Supplements Theorem Vertical Angles Vertical Angles Theorem Angle Addition Postulate Right Angles Theorem: Perpendicular lines intersect to form right angles. Right Angles Theorem: All right angles are congruent. Right Angles Theorem: Perpendicular lines form congruent adjacent angles. Right Angles Theorem: If two angles are congruent and supplementary, then each angle is a right angle. Right Angles Theorem: If two congruent angles form a linear pair, then they are right angles.

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19 Reasoning and Proof Test Review 1. What is the negation of the statement The Sun is shining? [1] It is cloudy. [3] It is not raining. [2] It is daytime. [4] The Sun is not shining 2. If S = "It is snowing." and C = "It is cold.", which statement is ~C V S? [1] It is not cold, and it is snowing. [2] If it is not cold, then it is snowing. [3] It is not snowing or it is cold. [4] It is not cold or it is snowing. 3. What is the converse of the statement: "If ABCD is a rectangle, then the diagonals bisect each other." [1] If the diagonals of ABCD bisect each other, then ABCD is a rectangle [2] If ABCD is not a rectangle, then the diagonals do not bisect each other. [3] ABCD is a rectangle and the diagonals bisect each other. [4] If the diagonals of ABCD do not bisect each other, then ABCD is not a rectangle. 4. Which statement is logically equivalent to If Andrea gets a job, she buys a new car? [1] Andrea gets a job and she buys a new car. [2] If Andrea does not buy a new car, she does not get a job. [3] If Andrea does not get a job, she does not buy a new car. [4] If Andrea buys a new car, she gets a job. 5. The sentence "If, then = 9." is TRUE. Which of the following statements could be used to fill the blank and maintain the truth value of the sentence? [1] 10/5 = 2 [2] 12/3 = 6 [3] both choice [1] and choice [2] could be used [4] neither choice [1] nor choice [2] could be used

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21 11. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Inverse: Converse: Contrapositive: Which one is logically equivalent to the original statement? Write a Two-Column Proof for each: 12. Given: JK = AB, KL = BC A B C Prove: JL = AC J K L

22 13. Given: 5x 3 = 2x Prove: x = - 9 / Given: B is the midpoint of AC. C is the midpoint of BD. Prove: AB = CD A B C D

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