Chapter 2 Review Flashcards
inductive reasoning
reasoning that uses a number of specific examples to arrive at a conclusion
conjecture
a concluding statement reached using inductive reasoning
counterexample
the opposite of an example; a false example
statement
a sentence that is either true or false, but not both
represented using a letter such as p or q
EX. p: A rectangle is a quadrilateral.
truth value
the truth (T) or falsity (F) of a statement
EX. p: A rectangle is a quadrilateral. (Truth value: T)
negation
a statement that has the opposite meaning and truth value of an original statment
not p is written ~p
compound statment
two or more statments joined by teh word and or or
conjunction
a compound statement using the word and
p and q is written p^q
*conjunctions are true ONLY if both statements are true
disjunction
a compound statment that uses the word or
p or q is written p ν q
truth table
a table used as a convenient method for organizing the truth values of statements
conditional statement
a statement that can be written in if-then form
EX. If mthen
if-then statement
if p then q is written p → q
hypothesis
the phrase immediately following the word if
represnted by p
conclusion
the phrase immediately following the word then
represented by q
related conditionals
statements that are based on a given conditional statement
p→q
converse
the statement formed by exchanging the hypothesis and conclusion of a conditional statement
p →q
EX. If
inverse
the statement formed by negating both the hypothesis and conclusion of the conditional
~p → ~q
EX. if mnot 35, then not an acute angle.
contrapositive
the statment formed by negating both the hypothesis and the conclusion of the converse of the conditional
~q → ~p
EX. If not an acute angle, then mnot 35.
logically equivalent
statements with the same truth values
EX. A conditional statement and its contrapositive are logically equivalent.
The converse and inverse of a conditional are logically equivalent.
deductive reasoning
Uses facts, rules, definitions, or properties to reach logical conclusions from given statements
Law of Detachment
If p → q is a true statement and p is true, then q is true.
EX.
Given:If a car is out of gas, then it will not start. Sarah’s car is out of gas.
Valid conclusion: Sarah’s car will not start.
Law of Syllogism
If p → q and q → r are true statements, then p → r is a true statement.
EX.
Given: If you get a job, then you will earn money. If you earn money, then you will buy a car.
Valid Conclusion: If you get a job, then you will buy a car.
postulate / axiom
a statement that is accepted as true without proof
Points and Lines Postulate 2.1
Through any two points, there is exactly one line
EX. Only one line goes through points A and B.
Ponits and Planes Postulate 2.2
Through any three noncollinear points, there is exactly one plane.
EX. Plane M is the only plane through noncollinear points A, B, and C
Points and Lines Postulate 2.3
A line contains at least two points.
Ex. The line contains points A and B.
Points and Planes Postulate 2.4
A plane contains at least three noncollinear points.
EX. Plane M contains points A, B, and C.
Points, Lines, and Planes Postulate 2.5
If two points lie in a plane, then the entire line containing those points lies in that plane.
EX. Points K and L are on the same plane.
Intersections of Lines Postulate 2.6
If two lines intersect, then their intersection is exactly one point.
EX. Lines l and m intersect at point A.
Intersections of Planes Postulate 2.7
If two planes intersect, then their intersection is a line.
EX. Planes F and G intersect in line w.
proof
a logical argument in which each statement is supported by a statement that is accepted as true
theorem
a statement of conjecture that has been proven
paragraph proof or informal proof
a paragraph used to explain why a conjecture for a given situation is true
two-column proof or formal proof
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.
Theorem 2.1: Midpoint Theorem
If M is the midpoint of AB, then AM ≅ BM
algebraic proof
a proof that is made up of a series of algebraic statements
Addition Property of Equality
If a = b, then a + c = b + c.
Subtraction Property of Equality
If a = b, then a - c = b - c.
Multiplication Property of Equality
If a = b, then a • c = b • c.
Division Property of Equality
If a = b and c ≠ 0, then a/C = b/C.
Reflexive Property of Equality
a = a
Symmetric Property of Equality
If a = b, then b = a.
Transitive Property of Equality
If a = b and b = c, then a = c.
Substitution Property of Equality
If a = b, then a may be replaced by b in any equation or expression.
Distributive Property
a(b+c) = ab + ac
Reflexive Property of Congruence (with Segments)
AB ≅ AB
Symmetric Property of Congruence (with Segments)
If AB ≅ CD, then CD ≅ AB
Transitive Property of Congruence (with Segments)
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
Postulate 2.11 Angle Addition Postulate
C is the interior of ∠BAD if and only if
m∠BAC + m∠CAD = m∠BAD
Theorem 2.3: Supplement Theorem
If two angles form a linear pair, then they are supplementary angles.
m∠1 + m∠2 = 180°
Theorem 2.4: Complement Theorem
If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.
m∠1 + m∠2 = 180
Reflexive Property of Congruence (with Angles)
∠1 ≅ ∠1
Symmetric Property of Congruence (with Angles)
If ∠1 ≅ ∠2, then ∠2 ≅ ∠1.
Transitive Property of Congruence (with Angles)
If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3.
Theorem 2.6: Congruent Supplements Theorem
Angles supplementary to the same angle or to congruent angles are congruent.
Theorem 2.7: Congruent Complements Theorem
Angles complementary to the same angle or to congruent angles are congruent.
Theorem 2.8: Vertical Angles Theorem
If two angles are vertical angles, then they are congruent.
EX. ∠1 ≅ ∠3 and ∠2 ≅ ∠4
Right Angle Theorem 2.9
Perpendicular lines intersect to form four right angles.
Right Angle Theorem 2.10
All right angles are congruent.
EX. If ∠1, ∠2, ∠3, and ∠4 are ∟, then ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4
Right Angle Theorem 2.11
Perpendicular lines form congruent adjacent angles.
EX: If line a is perpendicular to line b, then ∠1 ≅ ∠2, ∠2 ≅ ∠4, ∠3 ≅ ∠4 and ∠1 ≅ ∠3
Right Angle Theorem 2.12
If two angles are congruent and supplementary, then each angle is a right angle.
Right Angle Theorem 2.13
If two congruent angles form a linear pair, then they are right angles.
