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