Inductive reasoning

reasoning that uses a number of specific examples to arrive at a conclusion

Ex: 8:30a.m.-> 9:45a.m.-> 11:00a.m.-> 12:15a.m.

conjecture

a concluding statement reached using inductive reasoning

Ex: The show time is one hour and fifteen minutes greater than the previous show time. The next show will be 12:15p.m.+1:15 or 1:30

counterexample

a false example to how that a conjecture is not true

Ex: If n is a real number, then n(2)>n

COUNTEREXAMPLE: the conjecture is false, since 1(2)is not greater then 1

statement

a sentence that is either true or false

Ex: My dads car is green

truth value

that of a statement in which is true or false

Ex: A rectangle is a quadrilateral

negation

of a statement that has the opposite meaning, as well as the opposite value

Ex:a rectangle is not a quadrilateral

compound statement

two or more statements joined by the word and or or

Ex: A rectangle is a quadrilateral, and a rectangle is convex

conjunction

a compound statement using the word and; only true when both statements that form it are true

Ex: A rectangle is a quadrilateral, and a rectangle is convex

disjunction

a compound statement that uses the word or

Ex: Jesse studies Geometry, or Jesse studies chemistry

truth table

used to determine truth values of negations and compound statements

Ex: p -p

T F

F T

conditional statement

a statement that can be written in if-then form

Ex: If you would like to speak to a representative, then you will press 0 now

if-then statement

the form of if p, then q

Ex: If it rains, then we will not go outside

hypothesis

the phrase immediately following the word if

Ex: it rains

conclusion

the phrase immediately following the word then

Ex: we will not go outside

related conditionals

statements based on a given conditional statement

Ex: based on given conditional statements

converse

formed by exchanging the hypothesis and conclusion of the conditional

Ex: If <A is an acute angle, then m<A is 35

inverse

formed by negating both the hypothesis and conclusion

Ex: I m<A is not 35, then <A is not an acute angle

contrapositive

formed by negating both the hypothesis and conclusion of the converse of the conditional

Ex: If <A is not an acute angle, then m<A is not 35

logically equivalent

statements with the same truth values

Ex: If I wake up early, then I will take a shower

deductive reasoning

uses facts, rules definitions, or properties to reach logical conclusions from given statements

Ex: It is a complementary angle because the sides add up to 90 degrees

postulate

a statement that is accepted as true without proof

Ex: Through any two points, there is exactly one line

axiom

a statement that is accepted as true without proof

Ex: Through any two points, there is exactly one line

proof

a logical argument in which each statement you make is supported by a statement that is accepted as true

theorem

a statement or conjecture that has been proven; can be used as a reason to justify statements in other words

Ex: equation

deductive argument

formed by a logical chain of statements linking the given to what you are trying to prove

Ex: Given(Hypothesis)->Statements and Reasons->Prove(Conclusion)

paragraph proof

a paragraph explaining why a conjecture for a given statement is true

informal proof

a paragraph explaining why a conjecture for a given statement is true

algebraic proof

a proof that is made up of a series of algebraic statements

Ex: Distributive Property, Substitution Property, Division property

two-column proof

contains statements and reasons organized in two columns

Ex: Statements Reasons

C=13 Given

formal proof

contains statements and reasons organized in two columns

Ex: Statements Reasons

C=13 Given

Ruler Postulate

The points on any line or line segment can be put into one to one correspondence with real numbers.

Segment Addition Postulate

If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC.

Reflexive Property of Congruence

AB is congruent to AB

Symmetric Property of Congruence

If AB is congruent to CD, then CD is congruent to AB

Transitive Property of Congruence

If AB is congruent to CD and CD is congruent to EF, then AB is congruent to EF.

Protractor Postulate

Given any angle, the measure can be put into one to one correspondence with real numbers between the numbers 0 and 180

Angle Addition Postulate

m<ABD + m<DBC = m<ABC

Supplement Theorem

If two angles form a linear pair, then they are supplementary angles

Ex: m<1 + m<2 = 180

Complement Theorem

If the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles

Ex: m<1 + m<2 = 90

Congruent Supplements Theorem

Angles supplementary to the same angle or to congruent angles are congruent

Ex: If m<1 + m<2 = 180 and m<2 + m<3 = 180, then <1 is congruent to <3.

Congruent Complements Theorem

Angles complementary to the same angle or to congruent angles are congruent.

Ex: If <m4 + m<5 = 90 and m<5 + m<6 = 90, then <4 is congruent to <6

Vertical Angles Theorem

If two angles are vertical angles then they are congruent

Ex: <1 is congruent to <3 and <2 is congruent to <4

Right Angle Theorems

Perpendicular lines intersect to form four right angles

Ex: if AC is perpendicular to DB, then <1, <2, <3, and <4 are right angles.

Right Angle Theorems

All right angles are congruent

Ex: if <1, <2, <3, and <4 are right angles, then <1 is congruent to <2 is congruent to <3 is congruent to <4

Right Angle Theorems

Perpendicular lines form congruent adjacent angles

Ex: If AC is perpendicular to DB, then <1 is congruent to <2, <2 is congruent to <4, <3 is congruent to <4 and <1 is congruent to <3

Right Angle Theorems

If two angles are congruent and supplementary, then each angle is a right angle

Ex: if <5 is congruent to <6 and <5 is supplementary to <6, then <5 and <6 are right angles

Right Angle Theorems

If two congruent angles form a linear pair, then they are right angles

Ex: if <7 and <8 form a linear pair, then <7 and <8 are right angles