1 Geometry 2-1 Inductive Reasoning and Conjecturing A. Definitions 1. A conjecture is an guess. 2. Looking at several specific situations to arrive at a is called inductive reasoning. Ex 1: For points A, B, and C, AB = 10, BC = 8, and AC = 5. Make a conjecture and draw a figure to illustrate your conjecture. Given: Points A, B, and C AB = 10, BC = 8, and AC =5 Conjecture: 3. If a conjecture is made, and can be determined that it is false, it takes only one false example to show that a conjecture is not true. The false example is called a counterexample. Ex 2: Find a counterexample based on the given information. Given: points P, Q, and R are collinear Conjecture: Q is between P and R. Example 3: Based on the table showing unemployment rates for various counties in Kansas, find a counterexample for the following statement. The unemployment rate is highest in counties with the most people. County Civilian Labor Force Rate HW: Geometry 2-1 p odd, 43-44, odd Honors: Shawnee 90, % Jefferson 9, % Jackson 8, % Douglas 55, % Osage 10, % Wabaunsee 3, % Pottawatomie 11, %
2 Geometry Section Logic A. Determine Truth Values 1. Definitions a.) A statement is a sentence that is true or false, but not both. b.) The truth or falsity of a statement is called its truth value. c.) Statements are often represented by using the letters p and q. p: Visalia is a city in California. d.) The negation of a statement has the opposite meaning as well as the opposite truth value. Not p: Visalia is. (this could also be written ~p.) e.) Two or more statements can be joined to form a compound statement. p: Sacramento is a city in California. q: Sacramento is the capitol of California. The two statements can be joined by the word and. p and q: Sacramento is a city in California and Sacramento is the capital of California. f.) A conjunction is a compound statement formed by joining two or more statements with the word and. p q, read p and q. Example 1: Use the following information to write a compound statement for each conjunction. Then find its truth value. p: one foot is 14 inches q: September has 30 days. r: A plane is defined by 3 noncollinear points. a.) p and q b.) r and p
3 c.) ~ q r d.) ~ p r g.) A disjunction is a compound statement formed by joining two or more statements with the word or. 1. A disjunction is true if one or more of the statements are true. 2. p q is read p or q. Example 3: Use the following information to write compound statements for each disjunction. Then find its truth value. p: AB is the proper notation for line AB q: Centimeters are metric units. r: 9 is a prime number. a.) p or q b.) q r HW: Geometry 2-2 p all, 41-44, 54-55, odd Hon: 45-47
4 Section 2-3, If Then Statements and Postulates A. If-Then Statements 1. Statements can be changed into If-Then statements a. Football players are jocks. b. Smart kids are geeks. c. Boys are mean d. Adjacent angles have a common side. 2. If - Then statements are used to clarify confusing statements 3. If Then Statements are called conditional statements or conditionals. a. The part following the if is called the hypothesis. b. The part following the then is called the conclusion. c. p q represents the conditional statement if p, then q Example 1: Identify the hypothesis and conclusion of the following statements: a.) If this had been an actual emergency, then the attention signal you just heard would have been followed by official news, information or instruction. Hypothesis: Conclusion: b.) John will advance to the next level of play if he completes the maze on his computer game. Hypothesis: Conclusion Example 2: Write each statement in if-then form and identify the hypothesis and conclusion. a.) A five-sided polygon is a pentagon. B. Alterations of If-Then Statements 1. You can change an if-then statement by switching the hypothesis and conclusion. This new statement is called the converse. Example 2: Write the converse of the true conditional An angle that measures 120 o is obtuse. Step 1: Write the conditional in If-Then form. Step 2. Write the converse of the true conditional. Step 3. Determine if your new statement is true or false.
5 Step 4. Give a counterexample if it is not true. 2. The denial of a statement is called a negation. a. the negation of the statement an angle is acute is An angle is not acute b. : p represents not p or the negation of p 3. Given a conditional statement, the negation of both the hypothesis and conclusion is called the inverse. a. The inverse of p q is : p : q Example 3: Write the inverse of the true conditional Vertical angles are congruent. Determine if the inverse is true or false. If false, give a counterexample. If-Then form: Negate both the hypothesis and conclusion Determine if they are true or false 4. Given a conditional statement, its contrapositive can be formed by negating both the hypothesis and conclusion of the converse of the given statement. a. The contrapositive of p q is : q : p Example 4: Write the contrapositive of the true conditional If two angles are vertical, then they are congruent. Statement: Converse: Contrapositive: True or false? -The contrapositive of a true statement is always true and the contrapositive of a false statement is always false. HW Geometry 2-3 p all, all, 50-51, odd Hon: 46-47
6 Geometry 2-4 Deductive Reasoning A. Definitions 1. Deductive reasoning uses facts, rules, definitions, or properties to reach logical conclusions. 2. (Inductive reasoning uses examples to make a conjecture) 3. A form of deductive reasoning that is used to make conclusions from conditional statements is called the Law of Detachment. 4. The law of detachment says if p q is a true conditional, and there is a specific example where p is true, then q must be true. Example 1: Determine if the following conclusions are true from the Law of Detachment. a.) If you are at least 16 years old, you can get a drivers license. b.) Peter is 16 years old. c.) Peter can get a drivers license. a.) If you like pizza, you love Chicago style deep dish. b.) Jen loves Chicago style deep dish. c.) Jen likes pizza. 5. Another form of deductive reasoning is the law of syllogism, which states that: If p q is true, and q r is true, then p r must be true. Example 2: Determine if the following conclusions are true based on the law of syllogism. a.) If you are a careful bicycle rider, then you wear a helmet. b.) If you wear a helmet, then you look kind-of silly. c.) If you are a careful bicycle rider, then you look kind-of silly. a.) If you run every day, you will get healthier. b.) If you run every day, you will get faster. c.) If you get healthier, you will get faster.
7 Example 3: Determine whether statement 3.) follows from statements 1.) and 2.) by the law of syllogism or the law of detachment. If it does, state which law was used. If it does not, write invalid. 1.) If Sue wants to participate in the wrestling competition, she will have to meet an extra 3 times a week to practice. 2.) If Sue adds anything to her weekly schedule, she cannot take karate lessons. 3.) If Sue wants to participate in the wrestling competition, she cannot take karate lessons. 1.) If the girls want to win the volleyball game, they will have to play better. 2.) Shania is playing better. 3.) The girls will win the volleyball game. 1.) If you want to get an A in Geometry, you will have to study hard. 2.) Faith is studying hard. 3.) Faith will get an A in Geometry. 1.) If two angles are vertical, then they are congruent. 2.) 1 and 2 are vertical angles. 3.) 1 and 2 are congruent HW: Geometry 2-4 p all, 34, 35, odd Hon: 32
8 Geometry 2-5 Postulates and Paragraph Proofs A. Points lines and planes 1. In geometry, a postulate is a statement that describes a fundamental relationship between the basic terms of geometry. 2. Postulates are accepted as true. a.) Postulate 2-1 Through any two points, there is exactly 1 line. b.) Postulate 2-2 Through any three points not on the same line, there is exactly one plane. Example 1: Some snow crystals are shaped like regular hexagons. How many lines must be drawn to interconnect all vertices of a hexagonal snow crystal. c.) Postulate 2-3 A line contains at least two points d.) Postulate 2-4 A plane contains at least three points not on the same line. e.) Postulate 2-5 If two points lie in a plane, then the entire line containing those two points lies in that plane. f.) Postulate 2-6 If two lines intersect, then their intersection is exactly one point. g.) Postulate 2-7 If two planes intersect, then their intersection is a line. Example 2: Determine if each statement is always, sometimes, or never true. suur suur a. If plane T contains EF, and EF contains point G, then plane T contains point G. suur b. For XY, if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R. c. GH suur contains three noncollinear points. B. Paragraph proofs 1. Once a statement or conjecture has been proven to be true, it is called a theorem. 2. A proof is a logical argument in which each statement you make is supported by a statement that is accepted as true. a. One type of proof is called a paragraph proof, where you write a paragraph to explain why a conjecture for a given situation is true.
9 3. Proof writing -There are five parts to writing a good proof. a. State the theorem to be proved. b. List the given information. c. If possible, draw a diagram to illustrate the given information. d. State what is to be proved. e. Develop a system of deductive reasoning. Example 3: Given suur AC intersecting CD suur, write a paragraph proof to show ACD is a plane. suur suur AC and CD must intersect because if 2 lines intersect, then. Point A is on suur AC, and. Therefore, points are not. Therefore, ACD because it contains 3 points that are not. 4. Theorem Midpoint Theorem - If M is is the midpoint of AB, then AM MB. HW: Geometry 2-5 p all, all
10 Geometry 2-6 Algebraic Proofs A. Algebraic Proofs 1. Properties of Equality Properties of Equality for Real Numbers Reflexive Property For every number a, a = a. Symmetric Property For all numbers a and b, if a = b, then b = a. Transitive Property For all numbers a, b, and c, if a = b and b = c, then a = c. Addition and Subtraction Properties For all numbers a, b, and c, if a = b, then a + c = b + c and a - c = b - c. Multiplication and Division Properties For all numbers a, b, and c, if a = b, then a b agc = bg c, and if c 0, then = c c Substitution Property For all numbers a and b, if a = b, then a may be replaced by b in any equation or expression. Distributive Property For all numbers a, b, and c, a(b+c) = ab + ac. Example 1: Give a reason for each statement. Statements a.) If AB + BC = CE + BC, then AB = CE b.) m ABC = m ABC b.) c.) If XY = PQ and XY = RS, then PQ = RS d.) If 1/3x = 5, then x = 15 e.) If 2x = 9, then x = 9/2. 2. These types of proofs are called two column proofs. Reasons a.) c.) d.) e.)
11 Example 2: Justify each step in solving 3x + 5 = 7. 2 Given: 3x + 5 = 7 2 Prove: x = 3 Statements 1. 3x + 5 = x = 2(7) x + 5 = x = 9 5. x = 3 Reasons B. Geometric Proof 1. We can apply these properties to geometric expressions. Property Segments Angles Reflexive PQ = PQ mπ1 = Symmetric If AB = CD, then If mπa = mπb, then mπb = mπa Transitive If GH = JK and JK = LM, then GH = LM Example 3: Justify the steps for the proof. If mπ1 = mπ2 and, then mπ1 = mπ3. Given: ABD and DBC are complimentary angles. Prove: ABC is a right angle. Statements Reasons 1.) ABD and DBC are 1.) complimentary. 2.) m ABD + m DBC = 90 2.) 3.) m ABD + m DBC = m ABC 3.) 4.) m ABC = 90 4.) 5.) ABC is a right angle 5.) HW: Geometry 2-6 p all, 26, 37-38, odd Hon: 27, A B D C
12 Geometry 2-7 Proving Segment Relationships A. Segment Addition 1. Postulate Ruler Postulate - the points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero, and B corresponds to a positive real number. 2. Postulate Segment Addition Postulate - If B is between A and C, then AB + BC = AC. A B C Example 1: Given: Points A, B, C, and D are collinear. Prove: AB = AD BD A B C D Proof: Statements Reasons 1. Points A, B, C, and D 1. are collinear. 2. AD = AB + BD AD BD = AB AB = AD BD Theorem Congruence of segments is reflexive, symmetric, and transitive. a.) Reflexive Property - AB AB b.) Symmetric Property - If AB CD, then CD AB. c.) Transitive Property - If AB CD, and CD EF, then AB EF.
13 Example 2: Given: AC DF, AB DE Prove: BC EF Proof: Statements Reasons 1. AC DF, AB DE AC = DF, AB = DE AC AB = DF DE AC = AB + BC 4. DF = DE + EF 5. AC AB = BC 5. DF DE = EF 6. DF DE = BC BC = EF BC EF 8. A B C D E F HW: Geometry 2-7 p all, 21, 29-30, all Hon: 19, 22
14 Geometry Proving Angle Relationships A. Supplementary and Complementary Angles 1. Postulate Angle Addition Postulate - If R is in the interior of PQS, then m PQR + m RQS = m PQS. The converse is also true: If m PQR + m RQS = m PQS, then R is in the interior of PQS. Q P R S Example 1: If m PQR = 2x + 12, m RQS = 3x 2, and m PQS = 50, find the value of x and m PQS. Q P R S 2. Theorem 2-3 Supplemental Theorem - If two angles form a linear pair, then they are supplementary. 3. Theorem Complement Theorem - If the noncommon sides of two adjacent angles form a right angle, then the angles are complimentary angles. Example 2: If 1 and 2 form a linear pair, and m 1 = 4x 5 and m 2 = 14x + 5, find x, and m Theorem Congruence of angles is reflexive, symmetric, and transitive a. Reflexive Property b. Symmetric Property - If 1 2, then 2 1. c. Transitive Property - If 1 2, and 2 3, then Theorem Angles supplementary to the same angle or to congruent angles are congruent. a. If m 1+ m 2 = 180, and m 2 + m 3 = 180, then
15 Example 3: Proof of Theorem 2-6 Given: 1 and 2 are supplementary 2 and 3 are supplementary 1 y = Prove: Proof: Statements Reasons 1. 1 and 2 are supplementary 1. Given 2 and 3 are supplementary 2. m 1+ m 2 = m 2 + m 3 = Substitution 4. m 1 = m Theorem Angles complementary To the same angle or to congruent angles are congruent. a. If m 1+ m 2 = 90 o, and m 2 + m 3 = 90 o, then Theorem If two angles are vertical angles, then they are congruent. Example 4: If 1 and 2 are vertical angles, and m 1 = d 32, and m 2 = 175 2d, find m 1 and m Theorem Perpendicular lines intersect to form four. 9. Theorem All right angles are congruent. 10. Theorem Perpendicular lines form congruent adjacent angles. 11. Theorem If 2 angles are congruent and supplementary, then each angles is a right angle. 12. If two congruent angles form a linear pair, then they are. HW: Geometry 2-8 p all, 39, 41, 44-45, all Hon: 38