Chapter 2 Midterm Review

By: Mary Zhuang, Amy Lu, Khushi Doshi, Sayuri Padmanabhan, and

Madison Shuffler

IntroductionWrite 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

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.

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 _______

• 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

For example: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 qT T

T F

F T

F F

p q p qT TT FF TF F

T

F

F

F

T

T

T

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 IslandAustraliaAustralia is the conjunction

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

“then”

2-3 Conditional StatementsTruth Table when given 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

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 q Joining the conditional and converse

It snows if and only if they cancel school

False

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 ProofsPostulate: a statement that describes a fundamental relationship between basic terms of geometry2.1 Through any __ points, there is exactly 1 line2.2 Through any 3 points not on the _______ line, there is exactly 1 plane2.3 A _____ contains at least 2 points2.4 A plane contains at least __ points not on the same line2.5 If 2 points lie in a plane, then the entire _____ containing those points lies in that plane2.6 If 2 lines intersect, then their intersection is a _____2.7 If 2 _______ intersect, then their intersection is a line

2same

line3

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 information2.) Draw a diagram to illustrate the given information (if possible)3.) Use deductive reasoning4.) State what is to be ______proved

2-5 Postulates and Proofs

Definition of Congruent segments:

Definition of congruent Angles:

Midpoint Theorem:If M is the _______ of , then

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 ProofsGiven: 6x + 2(x – 1) = 30Statements1.) 6x + 2(x-1) = 302.) 6x + 2x – 2 = 303.) __________4.) 8x – 2 + 2 = 30 + 25.) ________6.) 8x/8 = 32/87.) x = 4

Prove: x = 4Reasons1.) ______2.) __________ ________3.) Substitution4.) Addition Property5.) Substitution6.) Division Property7.) ____________

Given

PropertyDistributive

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: is between and if and only if

A B C

2-7 Proving Segment Relationships

Segment Congruence• Reflexive Property: • Symmetric Property: If , then • Transitive Property: If and , then

2-7 Proving Segment RelationshipsFor 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): is in the interior of iff

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

rightcomplementary

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

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

samecongruent

complementarycongruent

2-8 Proving Angle Relationships• Vertical Angles Theorem: If two angles are vertical

angles, then they are congruentRight 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

linear

congruent

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

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