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 5x + 3 – 3 = -27 – 3 5x = -30 5x/5 = -30/5 Division X = -6

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: “A straight line segment can be drawn to join any two points” (2.1 Postulate) “Any straight line segment can be extended indefinitely in a straight line.” (definition of line) “Given any straight line segment, a circle can be drawn having the segments as radius and one endpoints as center.” “All right angles are congruent.” (right angle theorem) “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 _______ 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 ____ true proved not

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 - 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 - Negation of statement n is: We are ____ in school True not

2-2 Logic Conjunction: compound statement using “and” - A conjunction is only true when all the statements in it are _____ For example: Iced tea is cold and the sky is blue – Truth value is _____ Disjunction: compound statement using “or” - A disjunction is true if at least one of the statements is true May has 31 days or there are 320 days in an year – Truth value is true true 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 T F p q p q T F T T F T F T 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 Continent, Island, and Australia is the disjunction Australia is the conjunction Continent Australia Island

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 _____ Symbols: p → q, “if p, then q”, or “p implies q” “then”

2-3 Conditional Statements Symbols Formed by Example Truth Value Conditional p → q Using the given hypothesis and conclusion If it snows, then they will cancel school True 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 Contrapositive “switch-not” ∼q → ∼p Negating the hypothesis and conclusion and switching them If they do not cancel school, then it does not snow Biconditional p 1 q Joining the conditional and converse It snows if and only if they cancel school Truth Table when given Conditional Statements:

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 __ points, there is exactly 1 line 2.2 Through any 3 points not on the _______ line, there is exactly 1 plane 2.3 A _____ contains at least 2 points 2.4 A plane contains at least __ points not on the same line 2.5 If 2 points lie in a plane, then the entire _____ containing those points lies in that plane 2.6 If 2 lines intersect, then their intersection is a _____ 2.7 If 2 _______ intersect, then their intersection is a line 2 same line 3 line point planes

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 4.) State what is to be ______ proved

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: If M is the _______ of 𝐴𝐵 , then 𝐴𝑀 ≅ 𝑀𝐵 midpoint

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.) __________ 4.) 8x – 2 + 2 = 30 + 2 5.) ________ 6.) 8x/8 = 32/8 7.) x = 4 Prove: x = 4 Reasons 1.) ______ 2.) __________ ________ 3.) Substitution 4.) Addition Property 5.) Substitution 6.) Division Property 7.) ____________ Given Distributive Property 8x – 2 = 30 8x = 32 Substitution

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 S Q

2-8 Proving Angle Relationships 2.3 Supplement Theorem: if two angles form a _______ pair, then they are _____________ angles 2.4 Complement Theorem: If the noncommon sides of two adjacent angles form a _____ angle, then the angles are _____________ angles linear supplementary right complementary

2-8 Proving Angle Relationships Theorem 2.5: Congruence of angles is reflexive, symmetric, and transitive ________ Property: ∠1≅∠1 Symmetric Property: If ∠1≅ ∠2, then ∠2≅ ∠1 ________ Property: If ∠1≅ ∠2 and ∠2≅ ∠3, then ∠1≅ ∠3 Reflexive Transitive

2-8 Proving Angle Relationships 2.6 Congruent Supplement Theorem: Angles supplementary to the _____ angle or to congruent angles are _________ If m∠1+𝑚∠2=180 and m∠2+𝑚∠3=180, then ∠1≅ ∠3 2.7 Congruent Complement Theorem: Angles _____________ to the same angle or to congruent angles are _________ If m∠1+𝑚∠2=90 and m∠2+𝑚∠3=90, then ∠1≅ ∠3 same congruent complementary congruent

2-8 Proving Angle Relationships Vertical Angles Theorem: If two angles are vertical angles, then they are congruent Right Angle Theorems: 2.9.1 ____________ lines intersect to form four right angles 2.10 All right angles are __________ 2.11 Perpendicular lines form congruent adjacent angles 2.12 If two angles are congruent and supplementary, then each angle is a right angle 2.13 If two congruent angles form a ______ pair, then they are right angles Perpendicular congruent linear

Credits http://en.wikipedia.org/wiki/Euclid http://www.regentsprep.org/Regents/math/geometry/GPB/theorems.htm Google Images Geometry textbook

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