2-1 Inductive Reasoning & Conjecture INDUCTIVE REASONING is reasoning that uses a number of specific examples to arrive at a conclusion When you assume an observed pattern will continue, you are using INDUCTIVE REASONING. 1

2-1 Inductive Reasoning & Conjecture A CONCLUSION reached using INDUCTIVE REASONING is called a CONJECTURE. 2

2-1 Inductive Reasoning & Conjecture Example 1 Write a conjecture that describes the pattern in each sequence. Use your conjecture to find the next term in the sequence. 3

2-1 Inductive Reasoning & Conjecture Example 1a What is the next term? 3, 6, 12, 24, Conjecture: Multiply each term by 2 to get the next term. The next term is 24 •2 = 48. 4

2-1 Inductive Reasoning & Conjecture Example 1b What is the next term? 2, 4, 12, 48, 240 Conjecture: To get a new term, multiply the previous number by the position of the new number. The next term is 240•6 = 1440. 5

2-1 Inductive Reasoning & Conjecture Example 1c Conjecture: The number of small triangles is the perfect squares. What is the next shape? ? 1 4 9 6

2-1 Inductive Reasoning & Conjecture   The next big triangle should have _____ little triangles. 9 4 1 7

2-1 Inductive Reasoning & Conjecture EX 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. 8

2-1 Inductive Reasoning & Conjecture EX 2a The sum of an odd number and an even number is __________. Conjecture: The sum of an odd number and an even number is ______________. an odd number Example: 1 +4 = 5 Example: 26 +47 = 73 9

2-1 Inductive Reasoning & Conjecture EX 2b For points L, M, & N, LM = 20, MN = 6, AND LN = 14. 20 L N M 14 6 10

2-1 Inductive Reasoning & Conjecture L N M Conjecture: N is between L and M. OR L, M, and N are collinear. 11

2-1 Inductive Reasoning & Conjecture Counterexample: An example that proves a statement false 12

2-1 Inductive Reasoning & Conjecture Give a counterexample to prove the statement false. If n is an integer, then 2n > n. One possible counterexample: n = – 8 because… 2(– 8) = – 16 and – 16 > – 8 is false!!! 13

2-1 Inductive Reasoning & Conjecture Assignment: p.93 – 96 (#14 – 30 evens, 40 – 44, 64 – 66 all) 14

2-3 Conditional Statements CONDITIONAL STATEMENT A statement that can be written in if-then form An example of a conditional statement: IF Portage wins the game tonight, THEN we’ll be sectional champs. 15

2-3 Conditional Statements HYPOTHESIS the part of a conditional statement immediately following the word IF CONCLUSION the part of a conditional statement immediately following the word THEN 16

2-3 Conditional Statements Example 1 Identify the hypothesis and conclusion of the conditional statement. a.) If a polygon has 6 sides, then it is a hexagon. Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon 17

2-3 Conditional Statements Example 1 (continued) b.) Joe will advance to the next round if he completes the maze in his computer game. Hypothesis: Joe completes the maze in his computer game Conclusion: he will advance to the next round 18

2-3 Conditional Statements Example 2 Write the statement in if-then form, Then identify the hypothesis and conclusion of each conditional statement. 19

2-3 Conditional Statements a.) A dog is Mrs. Lochmondy’s favorite animal. If-then form: If it is a dog, then it is Mrs. Lochmondy’s favorite animal. Hypothesis: it is a dog Conclusion: it is Mrs. Lochmondy’s favorite animal 20

2-3 Conditional Statements b.) A 5-sided polygon is a pentagon. If-then form: If it is a 5-sided polygon then it is a pentagon. Hypothesis: it is a 5-sided polygon Conclusion: it is a pentagon 21

2-3 Conditional Statements CONVERSE: The statement formed by exchanging the hypothesis and conclusion 22

2-3 Conditional Statements Example 3 Write the conditional and converse of the statement. Bats are mammals that can fly. Conditional: If it is bat, then it is a mammal that can fly. Converse: If it is a mammal that can fly, then it is a bat. 23

2-3 Conditional Statements Assignment: p.109-111(#18 – 30, evens 50, 52) For #50 & 52, only write the conditional & converse. 24

2-4 Deductive Reasoning Recall from section 2-1….. When you assume an observed pattern will continue, you are using INDUCTIVE REASONING. 25

2-4 Deductive Reasoning Deductive reasoning uses facts, rules, definitions, or properties to reach logical conclusions from given statements. 26

2-4 Deductive Reasoning EX 1 Determine whether each conclusion is based on inductive reasoning (a pattern) or deductive reasoning (facts). 27

2-4 Deductive Reasoning a.) In a small town where Tom lives, the month of April has had more rain than any other month for the past 4 years. Tom thinks that April will have the most rain this year. Answer: inductive reasoning – Tom is basing his reasoning on the pattern of the past 4 years. 28

2-4 Deductive Reasoning b.) Sondra learned from a metoerologist that if it is cloudy at night, it will not be as cold in the morning as if there were no clouds at night. Sondra knows it will be cloudy tonight, so she believes it will not be cold tomorrow morning. Answer: deductive reasoning ‒ Sondra is using facts that she has learned about clouds and temperature. 29

2-4 Deductive Reasoning c.) Susan works in the attendance office and knows that all of the students in a certain geometry class are male. Terry Smith is in that geometry class, so Susan knows that Terry Smith is male. Answer: deductive reasoning – Susan’s reasoning is based on a fact she acquired while working in the attendance office. 30

2-4 Deductive Reasoning d.) After seeing many people outside walking their dogs, Joe observed that every poodle was being walked by an elderly person. Joe reasoned that poodles are owned exclusively by elderly people. Answer: inductive reasoning – Joe made his conclusion based on a pattern he saw.

2-5 Postulates and Paragraph Proofs Postulate or axiom – a statement that is accepted as true without proof

2-5 Postulates and Paragraph Proofs Theorem ‒ A statement in geometry that has been proven

2-5 Postulates and Paragraph Proofs a logical argument in which each statement you make is supported by a statement accepted as true

2-5 Postulates and Paragraph Proofs Examples of Postulates KNOW THESE POSTULATES!!! 2.1 Through any 2 points there is exactly one line. 2.2 Through any 3 noncollinear points there is exactly one plane.

2-5 Postulates and Paragraph Proofs 2.3 A line contains at least 2 points. 2.4 A plane contains at least 3 noncollinear points. 2.5 If 2 points lie in a plane, then the entire line containing those 2 points lies in that plane.

2-5 Postulates and Paragraph Proofs 2.6 If 2 lines intersect, then they intersect in exactly one point. 2.7 If 2 planes intersect, then they intersect in a line.

2-5 Postulates and Paragraph Proofs Example 2 Use the diagram on page 126 for these next examples. Explain how the picture illustrates that each statement is true. Then state the postulate that can be used to show each statement is true.

2-5 Postulates and Paragraph Proofs a.) Points F and G lie in plane Q and on line m. Therefore, line m lies entirely in plane Q. Answer: Points F and G lie on line m, and the line lies in plane Q. (2.5) If 2 points lie in a plane, then the line containing the 2 points lies in the plane.

2-5 Postulates and Paragraph Proofs b.) Points A and C determine a line. Answer: Points A and C lie along an edge, the line that they determine. (2.1) Through any 2 points there is exactly one line.

2-5 Postulates and Paragraph Proofs c.) Lines s and t intersect at point D. Answer: Lines s and t of this lattice intersect at only one location point D. (2.6) If 2 lines intersect, then their intersection is exactly one point.

2-5 Postulates and Paragraph Proofs d.) Line m contains points F and G. Point E can also be on line m. Answer: The edge of the building is a straight line m. Points E, F, and G lie along this edge, so they line along line m. (2.3) A line contains at least 2 points.

2-5 Postulates and Paragraph Proofs Midpoint Theorem: If M is the midpoint of then .

2-5 Postulates and Paragraph Proofs Assignment: p.120(#10-15 all) p.129-130(#16-28 even, 34-40 even)

2-6 Algebraic Proof Day 1 See p 2-6 Algebraic Proof Day 1 See p.134 for the book’s way of explaining this. PROPERTY DESCRIPTION OF PROPERTY Addition Property of Equality Subtraction Property of Equality Add the same value to both sides of an equation. Subtract the same value from both sides of an equation.

2-6 Algebraic Proof Day 1 DESCRIPTION OF PROPERTY PROPERTY Multiplication Property of Equality Division Property of Equality Multiply both sides of an equation by the same nonzero value. Divide both sides of an equation by the same nonzero value.

2-6 Algebraic Proof Day 1 a = a If a =b, then b = a. PROPERTY DESCRIPTION OF PROPERTY Reflexive Property of Equality Symmetric Property of Equality a = a If a =b, then b = a.

2-6 Algebraic Proof Day 1 DESCRIPTION OF PROPERTY PROPERTY Transitive Property of Equality Substitution Property If a = b and b = c, then a = c. If a = b, then a can be substituted for b in any equation or expression.

DESCRIPTION OF PROPERTY Distributive Property 2-6 Algebraic Proof Day 1 PROPERTY DESCRIPTION OF PROPERTY Distributive Property a(b + c) = ab + ac

2-6 Algebraic Proof Day 1 EXAMPLES State the property that justifies each statement.

Division Property of Equality Multiplication Property of Equality 2-6 Algebraic Proof Day 1 EX 1 If 3x = 15, then x = 5. Answer: Division Property of Equality OR Multiplication Property of Equality  

Subtraction Property of Equality Addition Property of Equality 2-6 Algebraic Proof Day 1 EX 2 If 4x + 2 = 22, then 4x = 20. Answer: Subtraction Property of Equality OR Addition Property of Equality

Transitive Property of Equality 2-6 Algebraic Proof Day 1 EX 3 If 7a = 5y and 5y = 3p, then 7a = 3p. Answer: Transitive Property of Equality

Transitive Property of Equality 2-6 Algebraic Proof Day 1 EX 4 If and , then Answer: Transitive Property of Equality

Reflexive Property of Equality 2-6 Algebraic Proof Day 1 EX 5 Answer: Reflexive Property of Equality

Symmetric Property of Equality 2-6 Algebraic Proof Day 1 EX 6 If 3 = AB, then AB = 3. Answer: Symmetric Property of Equality

Substitution Property of Equality 2-6 Algebraic Proof Day 1 EX 7 If AB = 7.5 and CD = 7.5, then AB = CD. Answer: Substitution Property of Equality

2-6 Algebraic Proof Day 1 Assignment: p.137-141(#1-4,9-16,46-48,56-58)

2-6 Algebraic Proof Day 2 EX 8 Write a justification for each step. Prove: If 2(5 – 3a) – 4(a + 7) = 92, then a = ‒11.

2-6 Algebraic Proof Day 2 STATEMENTS REASONS 2(5 – 3a) – 4(a + 7) = 92 Given Distributive Prop

2-6 Algebraic Proof Day 2 STATEMENTS REASONS –10a – 18 = 92 +18 +18 +18 +18 Substitution Prop Addition Prop

2-6 Algebraic Proof Day 2 STATEMENTS REASONS –10a = 110 a = –11 Substitution Prop Division Prop Substitution Prop

2-6 Algebraic Proof Day 2 EX 9 Write a two-column proof to verify the conjecture. If 4(3x + 5) – 4x = 2(3x + 5) – 8, then x = –9.

2-6 Algebraic Proof Day 2 STATEMENTS REASONS 4(3x + 5) – 4x = 2(3x + 5) – 8 12x + 20 – 4x = 6x + 10 – 8 Given Distributive Prop

2-6 Algebraic Proof Day 2 STATEMENTS REASONS 8x + 20 = 6x + 2 –6x –6x Substitution Prop Subtraction Prop

2-6 Algebraic Proof Day 2 STATEMENTS REASONS 2x + 20 = 2 –20 –20 –20 –20 Substitution Prop Subtraction Prop

2-6 Algebraic Proof Day 2 STATEMENTS REASONS 2x = –18 x = –9 Substitution Prop Division Prop Substitution Prop

2-6 Algebraic Proof Day 2 EX 10 If the distance d an object travels is given by d = 20t + 5, the time t that the object travels is given by Write a two column proof to verify this conjecture.

2-6 Algebraic Proof Day 2 STATEMENTS REASONS d = 20t + 5 d – 5 = 20t Given Subtraction Prop

2-6 Algebraic Proof Day 2 STATEMENTS REASONS Division Property Symmetric Prop

2-6 Algebraic Proof Day 2 EX 11 Write a two-column proof to prove the conjecture. If , mB = 2mC, and mC = 45, then mA = 90.

2-6 Algebraic Proof Day 2 mB = 2mC mC = 45 STATEMENTS REASONS mA = mB Given Defn of ≅ s

2-6 Algebraic Proof Day 2 STATEMENTS REASONS mA = 2mC mA = 2 (45) Substitution Prop Substitution Prop Substitution Prop

2-6 Algebraic Proof Day 2 Assignment: p.137 – 139(#5 – 7,17 – 19)

2-6 Algebraic Proof Day 2

2-6 Algebraic Proof Day 2

2-6 Algebraic Proof Day 2

2-6 Algebraic Proof Day 2