Chapter 2 Midterm Review

1. Chapter 2 Midterm Review By: Mary Zhuang, Amy Lu, KhushiDoshi, Sayuri Padmanabhan, and Madison Shuffler

2. Introduction Write a two-column proof. Given: 2(3x – 4) + 11 = x – 27 Prove: x = -6

3. EuclidΕὐκλείδης meaning, “good glory”300 BCAlso 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.

4. 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)

5. 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

6. 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.

7. 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

8. 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 atleast one of the statements is true For example: May has 31 days or there are 320 days in an year – Truth value is true true true

9. 2-2 Logic • Truth tables: organized method for truth value of statements Fill in the last column of each truth table: Conjunction: Disjunction: T T F T F T F F

10. 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 Australia Continent Island

11. 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”

12. 2-3 Conditional Statements Truth Table when given Conditional Statements:

13. 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)

14. 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

15. 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

16. 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

17. 2-5 Postulates and Proofs Definition of Congruent segments: Definition of congruent Angles: Midpoint Theorem: If M is the _______ of , then midpoint

18. 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

19. 2-6 Algebraic Proofs Prove: x = 4 Reasons 1.) ______ 2.) __________ ________ 3.) Substitution 4.) Addition Property 5.) Substitution 6.) Division Property 7.) ____________ 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 Given Distributive Property 8x – 2 = 30 8x = 32 Substitution

20. 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

21. 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: is between and if and only if A B C

22. 2-7 Proving Segment Relationships Segment Congruence • Reflexive Property: • Symmetric Property: If , then • Transitive Property: If and , then

23. 2-7 Proving Segment Relationships For Example: Given: A, B, C, and D are collinear, in that order; AB=CD Prove: AC=BD

24. 2-8 Proving Angle Relationships • Addition Postulate (2.11): is in the interior of iff P R Q S

25. 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

26. 2-8 Proving Angle Relationships • Theorem 2.5: Congruence of angles is reflexive, symmetric, and transitive • ________ Property: • Symmetric Property: If , then • ________ Property: If and ,then Reflexive Transitive

27. 2-8 Proving Angle Relationships • 2.6 Congruent Supplement Theorem: Angles supplementary to the _____ angle or to congruent angles are _________ • If and , then • 2.7 Congruent Complement Theorem: Angles _____________ to the same angle or to congruent angles are _________ • If and then same congruent complementary congruent

28. 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

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

30. Jeopardy

31. 2-5 2-4 2-3 2-2 2-1 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