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54 Cards in this Set
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conjecture

an educated guess based on known information


counterexample

an example that shows that a conjecture is not true.


statement

any statement that is either true, or false, but not both.


truth value

the truth or falsity of a statement


negation of a statement

a statement that has the opposite meaning as well as an opposite truth value.
~p read "not p" 

compound statement

joins two or more statements


conjunction

a compound statement formed by joining two or more statements with the word "and"
p ⋀ q read "p and q" a conjunction is true only when both statements are true. 

disjunction

a compound statement formed by joining two or more statements with the word "or"
p ⋁ q read "p and q" a disjunction is true when at least one of the statements are true. 

truth table

a way to organize truth values of statements


conditional statement

a statement that can be written in "ifthen" form
p → q read "if p, then q" or "p imples q" 

hypothesis

phrase immediately following the word "if"


conclusion

phrase immediately following the word "then"


conditional statement

p → q


converse statement

q → p


inverse statement

~p → ~q


contrapositive statement

~q → ~p


logically equivalent statement

conditional statement with the same truth values.


biconditional statement

the conjunction of a conditional and its converse.
(p → q) ⋀ (q → p) = p ↔ q read: "if and only if" 

inductive reasoning

reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction


deductive reasoning

the process that uses facts, rules, definitions, or properties to reach logical conclusions.


Law of Detachment

If p → q is true and p is true, then q is true.
Symbols: [(p → q) ∧ p] → q 

Law of Syllogism

If p → q and q → r are true, then p → r is true.
Symbols: [(p → q) ∧ (q → r)] → (p → r) 

Reflexive Property

For every real number a, a = a.


Symmetric Property

For all real numbers a and b, if a = b,
then b = a. 

Transitive Property

For all real numbers a, b, and c, if a = b and b = c, then a = c.


Addition Property

For all real numbers a, b, c, if a = b,
then a + c = b + c. 

Subtraction Property

For all real numbers a, b, c, if a = b,
then a  c = b  c. 

Multiplication Property

For all real numbers a, b, and c, ac = bc.


Division Property

For all real numbers a, b, and c, if a = b and c ≠ 0,
then a ÷ c = b ÷ c. 

Substitution Property

For all real numbers a and b, if a = b then a may be replaced by b in any equation or expression.


Distributive Property

For all real numbers a, b, and c,
a(b + c) = ab + ac. "Rainbow Rule" 

Commutative Property

For all real numbers a and b,
a + b = b + a (for addition), and ab = ba (for multiplication) 

Associative Property

For all real numbers a, b, and c,
a + (b + c) = (a + b) + c (for addition), and a(bc) = (ab)c (for multiplication) 

deductive proof

a group of algebraic steps used to solve a problem


formal proof
or two column proof 
statements and reasons justifying each statement organized in two columns


postulate
or axiom 
a statement that describes fundamental relationship between the basic terms of geometry


line

Through any two points, there is exactly one line.
A line contains at least two points. 

plane

Through any three points not on the same line, there is exactly one plane.
A plane contains at least three points not on the same line. 

If two points lie in a plane, then...

...the entire line containing those points lies in that plane.


If two lines intersect, then...

...their intersection is exactly one point.


If two planes intersect, then...

...their intersection is a line.


theorem

a statement or conjecture that has shown to be true


proof

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


Midpoint Theorem

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

Ruler Postulate

The points on any line or line segment 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.


Segment Addition Postulate

If B is between A and C,
then AB + BC = AC. 

Protractor Postulate

Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB, such that the measure of the angle formed is r.


Angle Addition Postulate

R is in the interior of ∠PQS, if and only if m∠PQR + m∠RQS = m∠PQS.


Supplement Theorem

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


Complement Theorem

If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.


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

... congruent.


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

... congruent.


Vertical Angles Theorem

If two angles are vertical angles, then they are congruent.


Right Angles

Perpendicular lines intersect to form four right angles.
All right angles are congruent. Perpendicular lines form congruent adjacent angles. If two angles are congruent and supplementary, then each angle is a right angle. If two congruent angles for a linear pair, then they are right angles. 