Chapter 2 Review Flashcards by Diana Law

inductive reasoning

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

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conjecture

a concluding statement reached using inductive reasoning

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counterexample

the opposite of an example; a false example

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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.

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truth value

the truth (T) or falsity (F) of a statement

EX. p: A rectangle is a quadrilateral. (Truth value: T)

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negation

a statement that has the opposite meaning and truth value of an original statment

not p is written _~p

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compound statment

two or more statments joined by teh word and or or

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conjunction

a compound statement using the word and

p and q is written p^q

*conjunctions are true ONLY if both statements are true

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disjunction

a compound statment that uses the word or

p or q is written p ν q

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truth table

a table used as a convenient method for organizing the truth values of statements

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conditional statement

a statement that can be written in if-then form

EX. If mthen

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if-then statement

if p then q is written p → q

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hypothesis

the phrase immediately following the word if

represnted by p

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conclusion

the phrase immediately following the word then

represented by q

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related conditionals

statements that are based on a given conditional statement

p→q

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converse

the statement formed by exchanging the hypothesis and conclusion of a conditional statement

p →q

EX. If

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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.

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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.

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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.

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deductive reasoning

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

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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.

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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.

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postulate / axiom

a statement that is accepted as true without proof

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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.

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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.