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Chapter 2 Midterm Review By: Mary Zhuang, Amy Lu, Khushi Doshi, Sayuri Padmanabhan, and Madison Shuffler Introduction Write a two-column proof. Given: 2(3x – 4) + 11 = x – 27 Prove: x = -6 Statement Reason 2(3x – 4) + 11 = x – 27 Given 6x – 8 + 11 = x – 27 Distributive 6x + 3 = x – 27 Substitution 6x – x + 3 = x – x – 27 Subtraction 5x + 3 = -27 Substitution 5x + 3 – 3 = -27 – 3 Subtraction 5x = -30 Substitution 5x/5 = -30/5 Division X = -6 Substitution Euclid Εὐκλείδης meaning, “good glory” 300 BC Also know as Euclid of Alexandria • Only a couple references that referred to him, nothing much is known about him and his life. • Known as the “father of geometry” • Created a book called The Elements, one of the best works for the history of mathematics • The Elements serves as the main textbook for mathematics, especially geometry. And that is where “Euclid Geometry” came from, which is what we learn today. How does Euclid relate to Chapter 2? Euclid actually created five postulates when he was alive, and we are introduced to postulates in Chapter 2. His five postulates are: 1. “A straight line segment can be drawn to join any two points” (2.1 Postulate) 2. “Any straight line segment can be extended indefinitely in a straight line.” (definition of line) 3. “Given any straight line segment, a circle can be drawn having the segments as radius and one endpoints as center.” 4. “All right angles are congruent.” (right angle theorem) 5. “If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less that two right angles, then the two lines inevitable must intersect each other on that side if extended far enough.” (parallel postulate) 2-1 Inductive Reasoning and Conjectures • Conjecture: An statement based on known information that is believed to be true but not yet proved _______ • Inductive reasoning: Reasoning that uses a number of specific examples or observations to arrive at a plausible generalization • Deductive reasoning: Reasoning that uses facts, rules, definitions, and/or properties to arrive at a conclusion • Counterexample: Example used to prove that a conjecture is ____ not true 2-1 Inductive Reasoning and Conjectures For example: If we are given information on the quantity and formation of the first 3 sections of stars, make a conjecture on what the next section of stars would be. 2-2 Logic • Statement: sentence that must be either true or false - Statement n: We are in school • Truth Value: whether the statement is true or false True - Truth value of statement n is _______ • Compound Statement: two or more statements joined: - We are in school and we are in math class • Negation: opposite meaning of a statement and the truth value, it can be either true or false not in school - Negation of statement n is: We are ____ 2-2 Logic • Conjunction: compound statement using “and” - A conjunction is only true when all the statements in it are _____ true For example: Iced tea is cold and the sky is blue – Truth value is _____ true • Disjunction: compound statement using “or” - A disjunction is true if at least one of the statements is true For example: May has 31 days or there are 320 days in an year – Truth value is true 2-2 Logic • Truth tables: organized method for truth value of statements Fill in the last column of each truth table: Conjunction: Disjunction: p q p q p q p q T T T T T T T F F T F T F T F F T T F F F F F F 2-2 Logic • Venn diagram - The center of the Venn diagram is the conjunction, also called the “and” statement - All the circles together make up the disjunction, also called the “or” statement Australia is the conjunction Continent Australia Island Continent, Island, and Australia is the disjunction 2-3 Conditional Statements • Conditional Statement: Statement that can be written in if-then form • Hypothesis: Phrase after the word “if” • Conclusion: Phrase after the word “then” _____ • Symbols: p → q, “if p, then q”, or “p implies q” 2-3 Conditional Statements Symbols Formed by Example hypothesis and conclusion they will cancel school Truth Value Truth pTable when given Conditional Statements: →q Using the given If it snows, then True Conditional Converse “switch” q→p Exchanging the hypothesis and conclusion If they cancel school, then it snows False Inverse “not” ∼p → ∼q Replacing the hypothesis and conclusion with its negation If it does not snow, then they will not cancel school False Contrapositive “switch-not” ∼q → ∼p Negating the hypothesis and conclusion and switching them If they do not cancel school, then it does not snow True Biconditional p Joining the conditional and converse It snows if and only if they cancel school False 1 q 2-4 Deductive reasoning • Law of Detachment: If p then q is true and p is true then, q is true. - Symbols: [(p→q) p]→ q • Law of Syllogism: If p then q and q then r are true, then p then r is also true. - Symbols: [(p→q) (q→r)]→(p→r) 2-5 Postulates and Proofs Postulate: a statement that describes a fundamental relationship between basic terms of geometry 2.1 Through any __ 2 points, there is exactly 1 line 2.2 Through any 3 points not on the _______ same line, there is exactly 1 plane 2.3 A _____ line contains at least 2 points 2.4 A plane contains at least __ 3 points not on the same line 2.5 If 2 points lie in a plane, then the entire _____ line containing those points lies in that plane point 2.6 If 2 lines intersect, then their intersection is a _____ 2.7 If 2 _______ planes intersect, then their intersection is a line 2-5 Postulates and Proofs • Theorem: A statement or conjecture shown to be true • Proof: A logical argument in which each statement you make is supported by a statement that is accepted as true • Two-column proof: a formal proof that contains statements and reasons organized in two columns. Each step is called a statement and the properties that justify each step are called ________ reasons 2-5 Postulates and Proofs Steps to a good proof: 1.) List the given information 2.) Draw a diagram to illustrate the given information (if possible) 3.) Use deductive reasoning proved 4.) State what is to be ______ 2-5 Postulates and Proofs Definition of Congruent segments: 𝐴𝑀 = 𝑀𝐵 ↔ 𝐴𝑀 ≅ 𝐵𝑀 Definition of congruent Angles: 𝑚∠𝐴 = 𝑚∠𝐵 ↔ ∠𝐴 ≅ ∠𝐵 Midpoint Theorem: midpoint of 𝐴𝐵, then 𝐴𝑀 ≅ 𝑀𝐵 If M is the _______ 2-6 Algebraic Proofs • The properties of equality can be used to justify each step when solving an equation • A group of algebraic steps used to solve problems form a deductive argument 2-6 Algebraic Proofs Given: 6x + 2(x – 1) = 30 Statements 1.) 6x + 2(x-1) = 30 2.) 6x + 2x – 2 = 30 3.) __________ 8x – 2 = 30 4.) 8x – 2 + 2 = 30 + 2 5.) ________ 8x = 32 6.) 8x/8 = 32/8 7.) x = 4 Prove: x = 4 Reasons 1.) ______ Given 2.) __________ Distributive ________ Property 3.) Substitution 4.) Addition Property 5.) Substitution 6.) Division Property Substitution 7.) ____________ 2-6 Algebraic Proofs • Since geometry also uses variables, numbers, and operations, many of the properties of equality used in algebra are also true in geometry 2-7 Proving Segment Relationships • Ruler Postulate: The points on any line 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. (This postulate establishes a number line on any line) • Segment Addition Postulate: 𝐵 is between 𝐴 and 𝐶 if and only if 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 A B C 2-7 Proving Segment Relationships Segment Congruence • Reflexive Property: 𝐴𝐵 ≅ 𝐴𝐵 • Symmetric Property: If 𝐴𝐵 ≅ 𝐶𝐷, then 𝐶𝐷 ≅ 𝐴𝐵 • Transitive Property: If 𝐴𝐵 ≅ 𝐶𝐷 and 𝐶𝐷 ≅ 𝐸𝐹, then 𝐴𝐵 ≅ 𝐸𝐹 2-7 Proving Segment Relationships For Example: Given: A, B, C, and D are collinear, in that order; AB=CD Prove: AC=BD 2-8 Proving Angle Relationships • Addition Postulate (2.11): 𝑅 is in the interior of ∠𝑃𝑄𝑆 iff 𝑚∠𝑃𝑄𝑅 + 𝑚∠𝑅𝑄𝑆 = 𝑚∠𝑃𝑄𝑆 P R Q S 2-8 Proving Angle Relationships • 2.3 Supplement Theorem: if two angles form a _______ linear pair, then they are _____________ supplementary angles • 2.4 Complement Theorem: If the noncommon right sides of two adjacent angles form a _____ angle, then the angles are _____________ complementary angles 2-8 Proving Angle Relationships • Theorem 2.5: Congruence of angles is reflexive, symmetric, and transitive Reflexive Property: ∠1 ≅ ∠1 • ________ • Symmetric Property: If ∠1 ≅ ∠2, then ∠2 ≅ ∠1 • ________ Transitive Property: If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3 2-8 Proving Angle Relationships • 2.6 Congruent Supplement Theorem: Angles same angle or to supplementary to the _____ congruent angles are _________ congruent • If m∠1 + 𝑚∠2 = 180 and m∠2 + 𝑚∠3 = 180, then ∠1 ≅ ∠3 • 2.7 Congruent Complement Theorem: Angles complementary _____________ to the same angle or to congruent angles are congruent _________ • If m∠1 + 𝑚∠2 = 90 and m∠2 + 𝑚∠3 = 90, then ∠1 ≅ ∠3 2-8 Proving Angle Relationships • Vertical Angles Theorem: If two angles are vertical angles, then they are congruent Right Angle Theorems: • 2.9.1 ____________ Perpendicular lines intersect to form four right angles • 2.10 All right angles are __________ congruent • 2.11 Perpendicular lines form congruent adjacent angles • 2.12 If two angles are congruent and supplementary, then each angle is a right angle linear pair, then • 2.13 If two congruent angles form a ______ they are right angles Credits • http://en.wikipedia.org/wiki/Euclid • http://www.regentsprep.org/Regents/math/ge ometry/GPB/theorems.htm • http://www.regentsprep.org/Regents/math/ge ometry/GPB/theorems.htm • Google Images • Geometry textbook Jeopardy 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 50 50 50 50 50 50 50 50