Geometry Chapter 2: Reasoning And Proof

Front 1

conjecture

Back 1

an educated guess based on known information

Front 2

counterexample

Back 2

an example that shows that a conjecture is not true.

Front 3

statement

Back 3

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

Front 4

truth value

Back 4

the truth or falsity of a statement

Front 5

negation of a statement

Back 5

a statement that has the opposite meaning as well as an opposite truth value.

~p read "not p"

Front 6

compound statement

Back 6

joins two or more statements

Front 7

conjunction

Back 7

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.

Front 8

disjunction

Back 8

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.

Front 9

truth table

Back 9

a way to organize truth values of statements

Front 10

conditional statement

Back 10

a statement that can be written in "if-then" form

p → q read "if p, then q" or "p imples q"

Front 11

hypothesis

Back 11

phrase immediately following the word "if"

Front 12

conclusion

Back 12

phrase immediately following the word "then"

Front 13

conditional statement

Back 13

p → q

Front 14

converse statement

Back 14

q → p

Front 15

inverse statement

Back 15

~p → ~q

Front 16

contrapositive statement

Back 16

~q → ~p

Front 17

logically equivalent statement

Back 17

conditional statement with the same truth values.

Front 18

biconditional statement

Back 18

the conjunction of a conditional and its converse.

(p → q) ⋀ (q → p) = p ↔ q read: "if and only if"

Front 19

inductive reasoning

Back 19

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

Front 20

deductive reasoning

Back 20

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

Front 21

Law of Detachment

Back 21

If p → q is true and p is true, then q is true.

Symbols: [(p → q) ∧ p] → q

Front 22

Law of Syllogism

Back 22

If p → q and q → r are true, then p → r is true.

Symbols: [(p → q) ∧ (q → r)] → (p → r)

Front 23

Reflexive Property

Back 23

For every real number a, a = a.

Front 24

Symmetric Property

Back 24

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

Front 25

Transitive Property

Back 25

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

Front 26

Addition Property

Back 26

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

Front 27

Subtraction Property

Back 27

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

Front 28

Multiplication Property

Back 28

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

Front 29

Division Property

Back 29

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

Front 30

Substitution Property

Back 30

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

Front 31

Distributive Property

Back 31

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

"Rainbow Rule"

Front 32

Commutative Property

Back 32

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

Front 33

Associative Property

Back 33

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

Front 34

deductive proof

Back 34

a group of algebraic steps used to solve a problem

Front 35

formal proof
or
two column proof

Back 35

statements and reasons justifying each statement organized in two columns

Front 36

postulate
or
axiom

Back 36

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

Front 37

line

Back 37

Through any two points, there is exactly one line.

A line contains at least two points.

Front 38

plane

Back 38

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.

Front 39

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

Back 39

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

Front 40

If two lines intersect, then...

Back 40

...their intersection is exactly one point.

Front 41

If two planes intersect, then...

Back 41

...their intersection is a line.

Front 42

theorem

Back 42

a statement or conjecture that has shown to be true

Front 43

proof

Back 43

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

Front 44

Midpoint Theorem

Back 44

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

Front 45

Ruler Postulate

Back 45

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.

Front 46

Segment Addition Postulate

Back 46

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

Front 47

Protractor Postulate

Back 47

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.

Front 48

Angle Addition Postulate

Back 48

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

Front 49

Supplement Theorem

Back 49

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

Front 50

Complement Theorem

Back 50

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

Front 51

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

Back 51

... congruent.

Front 52

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

Back 52

... congruent.

Front 53

Vertical Angles Theorem

Back 53

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

Front 54

Right Angles

Back 54

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.