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54 Cards in this Set
- Front
- Back
conjecture
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an educated guess based on known information
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counterexample
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an example that shows that a conjecture is not true.
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statement
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any statement that is either true, or false, but not both.
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truth value
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the truth or falsity of a statement
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negation of a statement
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a statement that has the opposite meaning as well as an opposite truth value.
~p read "not p" |
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compound statement
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joins two or more statements
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conjunction
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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. |
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disjunction
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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. |
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truth table
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a way to organize truth values of statements
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conditional statement
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a statement that can be written in "if-then" form
p → q read "if p, then q" or "p imples q" |
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hypothesis
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phrase immediately following the word "if"
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conclusion
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phrase immediately following the word "then"
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conditional statement
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p → q
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converse statement
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q → p
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inverse statement
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~p → ~q
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contrapositive statement
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~q → ~p
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logically equivalent statement
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conditional statement with the same truth values.
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biconditional statement
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the conjunction of a conditional and its converse.
(p → q) ⋀ (q → p) = p ↔ q read: "if and only if" |
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inductive reasoning
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reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction
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deductive reasoning
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the process that uses facts, rules, definitions, or properties to reach logical conclusions.
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Law of Detachment
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If p → q is true and p is true, then q is true.
Symbols: [(p → q) ∧ p] → q |
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Law of Syllogism
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If p → q and q → r are true, then p → r is true.
Symbols: [(p → q) ∧ (q → r)] → (p → r) |
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Reflexive Property
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For every real number a, a = a.
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Symmetric Property
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For all real numbers a and b, if a = b,
then b = a. |
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Transitive Property
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For all real numbers a, b, and c, if a = b and b = c, then a = c.
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Addition Property
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For all real numbers a, b, c, if a = b,
then a + c = b + c. |
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Subtraction Property
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For all real numbers a, b, c, if a = b,
then a - c = b - c. |
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Multiplication Property
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For all real numbers a, b, and c, ac = bc.
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Division Property
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For all real numbers a, b, and c, if a = b and c ≠ 0,
then a ÷ c = b ÷ c. |
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Substitution Property
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For all real numbers a and b, if a = b then a may be replaced by b in any equation or expression.
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Distributive Property
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For all real numbers a, b, and c,
a(b + c) = ab + ac. "Rainbow Rule" |
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Commutative Property
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For all real numbers a and b,
a + b = b + a (for addition), and ab = ba (for multiplication) |
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Associative Property
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For all real numbers a, b, and c,
a + (b + c) = (a + b) + c (for addition), and a(bc) = (ab)c (for multiplication) |
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deductive proof
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a group of algebraic steps used to solve a problem
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formal proof
or two column proof |
statements and reasons justifying each statement organized in two columns
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postulate
or axiom |
a statement that describes fundamental relationship between the basic terms of geometry
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line
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Through any two points, there is exactly one line.
A line contains at least two points. |
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plane
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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. |
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If two points lie in a plane, then...
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...the entire line containing those points lies in that plane.
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If two lines intersect, then...
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...their intersection is exactly one point.
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If two planes intersect, then...
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...their intersection is a line.
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theorem
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a statement or conjecture that has shown to be true
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proof
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a logical argument in which each statement is supported by statement that is accepted as true
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Midpoint Theorem
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If M is the midpoint of AB,
then AM ≅ MB. |
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Ruler Postulate
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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.
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Segment Addition Postulate
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If B is between A and C,
then AB + BC = AC. |
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Protractor Postulate
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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.
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Angle Addition Postulate
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R is in the interior of ∠PQS, if and only if m∠PQR + m∠RQS = m∠PQS.
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Supplement Theorem
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If two angles form a linear pair, then they are supplementary.
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Complement Theorem
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If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.
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Angles supplementary to the same angle or to congruent angles are ...
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... congruent.
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Angles complementary to the same angle or to congruent angles are...
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... congruent.
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Vertical Angles Theorem
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If two angles are vertical angles, then they are congruent.
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Right Angles
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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. |