Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning:

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1 Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning: Conjecture: Advantages: can draw conclusions from limited information helps us to organize our thinking human beings know how to use inductive reasoning naturally Disadvantages: sometimes we draw the wrong conclusion from the information our conclusions are only as strong as the evidence that we choose to consideration in mathematics, just one example which contradicts our conclusion tells us that our conclusion is wrong cannot be used to prove something mathematically Ex: #1: Find the next item in each pattern. January, March, May,... 7, 14, 21, 28,... Ex: #2: Complete each conjecture. The sum of two positive numbers is. The quotient of one positive number and one negative number is. The number of lines formed by four points, no three of which are collinear, is. If the side length of a square is doubled, the perimeter of the square is. Ex: #3: Biology Application A biologist observed blue-whale spouts of 25ft, 29ft, 27ft, and 24ft. Another biologist recorded humpback-whale spouts of 8ft, 7ft, 8ft, and 9ft. Make a conjecture based on the data. Page 1

2 To show that a conjecture is true you must prove it for all possibilities. To show that a conjecture is false you only have to show one counterexample. Counterexample: Examples: Show that each conjecture is false by finding a counterexample. For every integer n, n³ is positive. Two complementary angles are not congruent. Based on the data in Example 4C, the monthly high temperature in Abilene is never below 90 F for two months in a row. Chapter 2 Lesson 2: Conditional Statements Conditional Statement Conditional Statement: Definition Symbols Venn Diagram Hypothesis: Conclusion: Ex: #1 Identify the hypothesis and conclusion of each statement. If today is Thanksgiving Day, then today is A number is a rational number if it is an integer. Thursday. If an angle measures 40º, then it is an acute angle. Ex: #2 Write a conditional statement from each of the following. An obtuse triangle has exactly one obtuse angle. Congruent segments have equal measures. Page 2

3 Birds Blue Jays Adjacent angles Linear pair Truth value: When is a conditional false? Ex: #3 Determine if each conditional is true. If false, give a counterexample. If this month is August, then next month is If two angles are acute, then they are congruent. September. If an even number greater than 2 is prime, then = 8. If a number is odd, then it is divisible by 3. Negation: Chapter 2 Lesson 3: Using Deductive Reasoning to Verify Conjectures Deductive Reasoning: Advantages: proves that a statement is always true follows a systematic pattern of rules Disadvantages: can be difficult to use a skill that can take a lot of practice Ex: #1 Is each conclusion a result of inductive or deductive reasoning? Explain your answer. There is a myth that you can balance an egg on its There is a myth that the Great Wall of China is the end only on the spring equinox. A person was able only man-made object visible from the Moon. The to balance an egg on July 8, September 21, and Great Wall is barely visible in photographs taken December 19. Therefore this myth is false. from 180 miles above the Earth. The moon is about 237,000 miles from the Earth. Therefore the myth cannot be true. Page 3

4 Law of Detachment If p q is a true statement and p is true, then q is true. In your own words: Law of Syllogism If p q and q r are both true statements, then p r is a true statement. In your own words: Ex: #2 Determine if each conjecture is valid by the Law of Detachment. Given: If the side lengths of a triangle are 5cm, Given: In the World Series, if a team wins four 12cm, and 13cm, then the area of the triangle is games, then the team wins the series. The Red Sox 30cm². The area of ΔPQR is 30cm². won four games in the 2004 World Series. Conjecture: The side lengths of ΔPQR are 5cm, Conjecture: The Red Sox won the 2004 World 12, cm, and 13cm. Series. Ex: #3 Determine if each conjecture is valid by the Law of Syllogism. Given: If a figure is a kite, then it is a Given: If a number is divisible by 2, then it is quadrilateral. If a figure is a quadrilateral, then it is even. If a number is even, then it is an integer. a polygon. Conjecture: If a number is an integer, then it is Conjecture: If a figure is a kite, then it is a divisible by 2. polygon. Ex: #4 Draw a conclusion from the given information. Given: If 2y = 4, then z = -1. If x + 3 = 12, Given: If the sum of the measures of two angles is then 2y = º, then the angles are supplementary. If two angles are supplementary, they are not angles of a x + 3 = 12. triangle. m A = 135º, and m B = 45º. Page 4

5 Chapter 2 Lesson 4: Biconditional Statements and Definitions Biconditional: Definition Symbol All mathematical definitions can be written as biconditional statements. An angle is obtuse if and only if its measure is greater than 90º and less than 180º. A figure is circle if and only if it is the set of all points that are the same distance from a given point. When is a biconditional true? Definition: Polygon: Triangle: Quadrilateral: Page 5

6 Chapter 2 Lesson 5: Algebraic Proofs Proof: Algebraic Properties of Equality For any real number a, b, and c: Addition Property Multiplication Property Reflexive Property Symmetric Property Transitive Property Substitution Property Distributive Property The solution to an algebraic equation is a type of proof. The steps must appear in the correct order, and you must be able to justify each step. Ex: #1 Solve the equation and write a justification for each step. 1. 4m 8 = -12 Given Page 6

7 1. n + 8 = -6 5 Given Ex: #3 Write a justification for each step. 4 x - A 3 x + 5 D N NO = NM + MO 2 x M 3 x - O B m A B C = 8 x 6 x C 4x 4 = 2x + (3x 9) 4x 4 = 5x 9-4 = x 9 5 = x m ABC =m ABD + m DBC 8x = (3x + 5) + (6x 16) 8x = 9x 11 -x = - 11 x = 11 Properties of Congruence Reflexive Property of Congruence: Symmetric Property of Congruence: Transitive Property of Congruence: Each property of congruence has a corresponding property of equality. Page 7

8 Ex: #4 Identify the property that justifies each statement. If EF = GH and GH = JK, then EF = JK AB = 14 then 14 = AB m EJM = m MJE If 2( x 5 ) = 7, then 2x 10 = 7 AB CD and CD EF, then AB EF QRS QRS Chapter 2 Lesson 6: Geometric Proof Theorem: Two-Column Proof: Theorem Hypothesis Conclusion Linear Pair Theorem: If two angles form a linear pair, then they are supplementary. Right Angle Congruence Theorem: all right angles are congruent. Ex: #1 Write a justification for each step, given that A and B are supplementary and m A = 45º Statements Reasons A and B are supplementary and m A = 45º m A + m B = m B = 180 m B = 135 Page 8

9 M Ex: #2 Given: N is the midpoint of MP, Q is the midpoint of RP and PQ MN. Prove: PN QR P N Q R Statements Reasons N is the midpoint of MP, Q is the midpoint of RP PQ MN PN MN PQ PN PQ PN QR QR Ex: #3 Fill in the blanks to complete the two-column proofs. Given: XY Prove: XY XY Statements Reasons Given XY = XY Def. of congruent segments. Ex: #4 Write a two-column proof. Given: 1 and 2 are supplementary and 2 and 3 are supplementary Prove: 1 3 Statements Reasons Page 9

10 E Ex: #5 Given: FD bisects EFC and FC bisects DFB Prove: EFD CFB F B C D Statements Reasons Page 10

11 Chapter 2 Extension: Introduction to Symbolic Logic Compound Statement: Truth Table: Compound Statements Term Words Symbols Example Conjunction Disjunction When is a conjunction true? When is a disjunction true? Examples: Use p, q, and r to write a compound statement, then find its truth value. p: the month after April is May q: the next prime number after 13 is 17 r: half of 19 is 9 p q q r Examples: Construct a truth table for the following compound statements. u v u v u v p q r p q (p q) r Page 11

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