Warm-Up: What CAN or cannot Be Assume? In Chap 1, you have used figures to help describe or demonstrate different relationships among points, segments, lines, rays, and angles. When provided with a figure, there are certain relationships that can be assumed and others that cannot be assumed. CAN vs CANNOT M N O L P Q

Inductive Reasoning & Conjecturing How to connect reasoning and proofs? Inductive Reasoning & Conjecturing

Discuss This type of Reasoning My trip to Bahamas My experience at McDonalds

WarmUp: Facts Complete each sentence 1. _____ points are points that lie on the same line. 2. _____ points are points that lie on the same plane. 3. The sum of the measures of two _____ angles is 90 degrees.

Video: The Basics of Geometry What is Geometry? Who were the Geometers who introduce mathematics? What is inductive reasoning and deductive reasoning?

Inductive Reasoning Definition: a reasoning in which you look at patterns/several facts and then make an educated guess based on those facts.

Conjecture The educated guess made based on facts is called a conjecture. Synonyms:

Counterexample: Examples that shows conjectures are false is called counter examples.

Practice: Given: Collinear points D, E, and F. Part I: Determine if each conjecture is true or false based on the given information. Explain your answer and give a counterexample for any false conjecture. Given: Collinear points D, E, and F. Conjecture: DE + EF = DF. Given: <A and <B are supplementary. Conjecture: <A and <B are adjacent angles. Given: <D and <F are supplementary. <E and <F are supplementary. Conjecture: <D = <E Given: AB is perpendicular to BC. Conjecture: <ABC is a right angle.  

Practice: YOU PARTNER Conjectures: Counterexample: 1. 1. 2. 2. 3. 3. Part II:  You have sent your BFF 6 text messages in the last hour, but he/she has not responded. Make a list of 5 conjectures as to why she has not responded.   YOU PARTNER Conjectures: Counterexample:  1. 1. 2. 2.    3. 3. 4. 4. 5. 5.

Practice: P. 77 #1- 10, 24- 27 skip 6

Warm-up: Inductive reasoning Determine if the conjecture is TRUE or FALSE based on the given info. Explain your answer and give a counterexample for any false conjecture. 1. GIVEN: <A & <B are supplementary. CONJECTURE: <A &<B are not congruent. 2. GIVEN: m<A > m<B; m<B > m<C CONJECTURE: m<A > m<C. 3. GIVEN: segments AB, BC, AC CONJECTURE: A, B, & C are collinear. 4. GIVEN: <A &<B are vertical angles. CONJECTURE: <A & <B are congruent.

√ Practice P. 77 # 1-10, 24 - 27

Conditional statements Chapter 2 Section 2 Write the converse, inverse and contrapositive of a conditional statement. Identify, write, and analyze the truth value of conditional statements Conditional statements

Conditional Statements: Coach’s Statement Parent’s Statement

IF-Then Statements can be used to CLARIFY statements that may seem confusing; gives a clear understanding. If hypothesis, then conclusion. Hypothesis = “P” Conclusion = “Q” If- then Statements Format->

If- then statement: P -> Q If hypothesis, then conclusion. Example: Given: Teenagers are younger than 20. Statement: If you are a teen, then you’re younger than 20.

Converse: Q -> P If conclusion, then hypothesis. Example: Given: Teenagers are younger than 20. Statement: If you’re younger than 20, then you are a teen.

Inverse: ~P -> ~Q If “not” hypothesis, then “not” conclusion. Example: Given: Teenagers are younger than 20. Statement: If you are not a teen, then you’re not younger than 20.

Contrapositive: ~Q -> ~P If “not” conclusion, then “not” hypothesis. Example: Given: Teenagers are younger than 20. Statement: If you’re not younger than 20, then you are not a teen.

Practice: Rewrite the following statement in if-then form Practice: Rewrite the following statement in if-then form. The write the converse, inverse, and contrapostive. All elephants are mammals. If-then form: Converse: Inverse: Contrapostive:

Practice: Match the hypothesis on the left with a conclusion at the right that makes a true conditional statement. 1. If two angles form a linear pair,   2. If two angles are vertical, 3. If two adjacent angels form a right angle, a. then the angles are congruent. b. then the angles are complementary. c. then the angles are supplementary.

1. If today is Monday, then tomorrow is Tuesday. Practice: Identify the hypothesis and conclusion of each conditional statement. 1. If today is Monday, then tomorrow is Tuesday. 2. If a truck weighs 2 tons, then it weighs 4000 pounds. Write each conditional statement in if-then form. 3. All chimpanzees love bananas. 4. Collinear points lie on the same line.

Practice: Write the converse, inverse, and contrapositive of each conditional. 5. If an animal is a fish, then it can swim. 6. All right angles are congruent.

Practice: P.84 # 9-11 and 19-21

Warm-UP: Conditional Statements 1. Identify the hypothesis and conclusion: If its a triangle, then it have three angles. 2. Write the converse and truth value: If its Thursday, then Octavia will go swimming. 3. Write the inverse and truth value : If its cold outside, then I will not wear shorts. 4. Write the contrapositive and truth value : If a number is divisible by 6, then it is divisible by 2.

√ Practice: Page 84-85 # 9-11, 19-21 Review Discussion Conditional Statements Inductive Reasoning

Chapter 2 Quiz pt 1 Take your time Read each question carefully Label if needed No talking Remember Geometry is a VISUAL subject!!

Chapter 2 Section 3 Deductive Reasoning

Deductive Reasoning What conclusion can be drawn? Basketball game pt 1

Deductive Reasoning Deductive reasoning- a system of reasoning used to reach conclusions that must be true whenever the assumptions on which the reasoning is based upon is true.

Inductive vs. Deductive Reasoning Compare and Contrast reasoning- Inductive Reasoning- Uses prior experiences, assumptions, & patterns to reach a conclusion Deductive Reasoning- Uses rules, laws & facts to reach a conclusion

2 Laws Of Logic Law of Detachment- EXAMPLE: If p -> q is a true statement, and p (hypothesis) is true, then q (conclusion) is true. If the measure is 90º, then it’s a right angle, < B is 90º Conclusion:

Two Laws of logic Law of Syllogism- EXAMPLE: If p -> q is true and q -> r is true, then p -> r is also true. If I’m eating, then I’m eating cookies. If I’m eating cookies, then I’m drinking milk. Conclusion:

Practice: 2-3 Deductive Reasoning 1. If Jim is a Texan, then he is American. Jim is a Texan Conclusion: 2. If Spot is a dog, then he has four legs. Spot has four legs. 3. If Rachel lives in Tampa, then she lives in FL. If Rachel lives in FL, then she lives in the US.

Practice: Deductive reasoning Complete in textbook page.91 # 1 - 8

Warm-Up: Fill-in-the-blanks 1. Law of Detachment- If ______, then _____. ______ . Conclusion: ______. 2. Law of Syllogism- If _____, then ______. If _____ then ……… Conclusion: If ___, then …....

Solve Logic Puzzles: Example 1: Pets Bonnie, Cally, Daphne, and Fiona own a bird, cat, dog, and fish.  CLUES: No girl has a type of pet that begins with the same letter as her name. Bonnie is allergic to animal fur. Daphne feeds Fiona’s bird when Fiona is away. Complete table to determine who owns which animal. Complete table to determine who owns which animal. Bird Cat Dog Fish Bonnie Cally Daphne Fiona

Solve Logic Puzzles: Example 2: School Dance Ally, Emily, Misha, and Tracy go to a dance with Danny, Frank, Jude, and Kian.   CLUES: Ally and Frank are siblings. Jude and Kian are roommates Misha does not know Kian. Emily goes with Kian’s roommate. Tracy goes with Ally’s brother.   Complete table to determine who went to the school dance with whom. Danny Frank Jude Kian Ally Emily Misha Tracy

Logic puzzle: object of the game- to use deductive reasoning to correctly determine the type and name of each student’s pet. Logic Puzzle- There are 5 students: Alex, Carol, Mark, Sean, Tamara. Each student has one pet: a dog, a cat, a fish, a guinea pig or a rabbit. The pets names are Buddy, Fang, Merlin, Prince, and Stripe. Alex Carol Mark Sean Tamara Buddy Fang Merlin Prince Stripe Dog Cat Fish Guinea Pig Rabbit

Logic Puzzle Clues: 1. Tamara has a cat. 2. The cat’s name is Stripe. Alex Carol Mark Sean Tamara Buddy Fang Merlin Prince Stripe Dog Cat Fish Guinea Pig Rabbit Clues: 1. Tamara has a cat. 2. The cat’s name is Stripe. 3. Carol’s pet is Buddy. 4. Sean does not have a dog. 5. The guinea pig is not owned by Carol, Mark or Sean. 6. Neither Mark or Sean has a fish. 7. If Alex has a guinea pig, then its name is Prince. 8. Fang is not the name of the dog.

Practice: Check homework page 91# 1-8 Take Quiz Part 2. Practice: Logic Puzzles

Warm-Up: Deductive Reasoning 1. Define: A. Deductive reasoning- B. Law of Detachment- C. Law of Syllogism 2. List the 9 properties of equalities. Sec 2-5.

√Quiz: part 2 Homework: Logic Puzzle

Properties of equality Read properties on page 104 Make A Foldable Fold Hot Dog style Measure every 1.25” Should make 9 sections

Properties of Equalities for Real Numbers Addition: Subtraction: Multiplication: Division: If a = b, then a + c = b + c. If a = b, then ac = bc. If a = b, then a/c = b/c. c cannot equal 0. If a = b, then a - c = b - c.

Properties of Equalities for Real Numbers Reflexive: Symmetric: Transitive: For all real numbers, a = a For all real numbers, if a = b and b = c then a = c For all real numbers, if a = b, then b = a

Properties of Equalities for Real Numbers Distributive: Substitution: For all real numbers, if a = b, then b can be substituted for a in any Expression. For all real numbers, if a(b + c), then ab + ac and if a(b - c), then ab – ac.

Properties of Equalities for Real Numbers Complete in textbook Page 107 # 1-6 & 10 -15

Warm-Up: List the property of each step 1. Solve for x. Given: 3(x + 5) – 18 = 21 Solve for x. Given: If y = 2, then 2yx + 3y2 = 44

Properties of Equalities for Real Numbers Complete in textbook Check Page 107 # 1-6, 10 -15

Chapter 2 study guide Test tomorrow Inductive Reasoning Conjectures and Counterexamples Conditional Statements and Truth Value Deductive Reasoning Law of Detachment and Law of Syllogism Properties of Equalities

Deductive Reasoning Reflexive Property Symmetric Property Transitive Property Addition Property Subtraction Property Multiplication/ Division Property Substitution Property Distributive Property Law of Detachment Law of Syllogism

Warm-up: reasoning and proof test today, yeah! 1. Determine if the conjecture is true. If not, write or draw a counterex. Given: Line k and Line m Conjecture: Line k and m intersect. 2. Write the contrapositive and give truth value. ~ If its 8:10, then school has started. 3. Write the law and the conclusion. (1) If my age is 18, then I can vote. (2) If I can vote, then I can voice my opinion. (3) Law: 4. Name Property- If 5x = 35, then x = 7.

Name the property of equality. If 4 + 1 = 5, then 5 = 4 + 1 Answer: Symmetric

Name the property of equality. If m < 1 – 60 = 90, then m< 1 = 150 Answer: Addition

Name the property of equality. For all numbers, 534,020 = 534,020 Answer: Reflexive

Name the property of equality. If 5x + 12 = 32, then 5x = 20 Answer: Subtraction

Name the property of equality. If 4x = 12, then x = 3. Answer: Division

Name the property of equality. If 2(3x + 6), then 2 · 3x + 2 · 6 Answer: Distributive

Name the property of equality. If 3x = 12 and 12 = x + 5, then 3x = x + 5. Answer: Transitive

Name the property of equality. If x/25 = 4, then x = 100. Answer: Multiplication

Name the property of equality. Answer: Substitution If x = y and x = 50, then y = 50.

Vocabulary Review- In 3’s Match vocabulary and definition Turn all cards face down. Flip two cards each time until you find a match(pair)

Ready for Test… During test No TALKING Take your time GOOD Luck!

Warm-Up: Proof Statement Reasons Fill in the blanks to complete a two column proof of the Linear Pair Theorem. Given: <1 & <2 form a linear pair. Prove: <1 & <2 are supplementary. 1 2 Proof: A B C Statement Reasons 1. <1 & <2 form a linear pair 1. Given 2. BA & BC form a line 2. Def. of lin. Pair 3. m<ABC = 180° 3. _________________ 4. __________________ 4. < Add. Post. 5. __________________ 5. Subst. 6. <1 & < 2 are supplementary 6. ___________________ Word Bank Def. of supplementary Def. of lin. Pair m<1 + m<2 = 180° m<1 + m<2 = m<ABC

Warm-Up: Proof (~= congruent) Fill in the blanks to complete a two column proof of the Congruent Supplements Thm. Given: <1 & <2 are supplementary. <2 & <3 are supplementary Prove: <1 ~= <3 Proof: Statement Reasons 1. __________________ 1. Given 2. m<1 + m<2 = 180° 2. ________________ m<2 + m<3 = 180° 3. __________________ 3. Substitution 4. __________________ 4. ________________ 5. m<1 = m<3 5. ________________ 6. __________________ 6. Def. of congruent angles Word Bank Subtraction Property of = Def. of supplementary Reflexive Property m<1 + m<2 = m<2 + m<3 < 1 ~= < 3 <1 + <2 are supp, <2 + <3 are supp m<2 = m<2 Figure on board