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1

conjecture

an educated guess based on known information and specific examples

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2

counterexample

an example that contradicts the conjecture showing that the conjecture is not always true; only one counterexample is needed to prove that the entire conjecture is false

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3

statement

any sentence that is either true or false, but not both

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4

truth value

the truth or falsity of a statement

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5

compound statement

two or more statements joined by the word "and" or "or"

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6

conjunction

a compound statement formed by joining two or more statements with the word and; a conjunction is true only when all of its statements are true; signaled by "^"

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7

disjunction

a compound statement formed by joining two or more statements with the word or; a disjunction is true if at least one of its statements are true; signaled by "ˇ"

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8

conditional statement

a compound statement that consists of a premise, or hypothesis, and a conclusion, which is false only when its premise is true and its conclusion is false

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9

if-then statement

a compound statement of the form "if p, then q" where p and q are statements; written as "p -> q", "if p, then q" and "p implies q"

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10

hypothesis

the phrase immediately following the word "if" in a conditional statement

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11

conclusion

the phrase immediately following the word "then" in a conditional statement

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12

converse

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

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13

inverse

the statement formed by negating both the hypothesis and conclusion of a conditional statement

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14

contrapositive

the statement formed by negating both the hypothesis and conclusion of the converse of a conditional statement

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15

biconditional statement

the conjunction of a conditional and its converse; written as either "(p -\> q)^(q -\>p) -\> (p <-\> q)", or "p if and only if q"; "if and only if" can also be abbreviated as "iff"

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16

inductive reasoning

the process of reaching a conclusion based on a pattern of examples; assuming that an observed pattern may continue

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17

deductive reasoning

using general facts, rules, definitions, or properties to reach specific valid conclusions from given statements

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18

valid argument

an argument is valid if it is impossible for all of the premises, or supporting statements, of the argument to be true and its conclusion false

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19

Law of Detachment

if p -> q is a true statement and p is true, then q is true; Example: If a car is out of gas (p), then it will not start (q). Sarah's car is out of gas (p is true). Therefore, Sarah's car will not start (q is true); related to deductive reasoning

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20

Law of Syllogism

if p ->q and q -> r are true statements, then p->r is a true statement; Example: If you get a job (p), then you will earn money (q). If you will earn money (q), then you will buy a car (r). Therefore, if you get a job (p), you will buy a car (r)

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21

Postulate 3.1 (the numbers of the postulates are not necessary to know)

through any two points, there is exactly one line

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22

Postulate 3.2

through any three noncollinear points, there is exactly one plane

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23

Postulate 3.3

a line contains at least two points

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Postulate 3.4

a plane contains at least three noncollinear points

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25

Postulate 3.5

if two points lie in a plane, then the entire line containing those points lies in that plane

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Postulate 3.6

if two lines intersect, then their intersection is exactly one point

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Postulate 3.7

if two planes intersect, then their intersection is a line.

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proof

a logical argument in which each statement you make is supported by a statement that is accepted as true; these supporting statements may include definitions, postulates, and theorems

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two-column proof

a proof in which the steps are written in the left column and the corresponding reasons are written in the right column.

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30

deductive argument

an argument that proves a statement by building a logical chain of statements and reasons

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flow proof

a type of proof that uses boxes and arrows to show the flow of a logical argument

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32

paragraph proof

a proof written in the form of a paragraph that explains why a conjecture for a given situation is true; includes the undefined terms, theorems, definitions, or postulates that support each statement

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33

Postulate 3.8: Ruler Postulate

the points on any line or line segment can be put into one-to-one correspondence with real numbers.

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34

Postulate 3.9: Segment Addition Postulate

if A, B, and C are collinear, then point B is between A and C if and only if AB+BC=AC

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Postulate 3.10: Protractor Postulate

every angle has a measure that is between 0 and 180

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Postulate 3.11: Angle Addition Postulate

D is in the interior of <ABC if and only if m<ABD + m<DBC = m<ABC

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37

Theorem 3.3: Supplement Theorem

if two angles form a linear pair, then they are supplementary angles

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38

Theorem 3.4: Complement Theorem

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

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39

Theorem 3.5: Reflexive Property of Congruence

any geometric object is congruent to itself

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40

Theorem 3.5: Symmetric Property of Congruence

if one geometric object is congruent to a second, then the second object is congruent to the first.

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41

Theorem 3.5: Transitive Property of Congruence

if one geometric object is congruent to a second, and the second is congruent to a third, then the first object is congruent to the third object.

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42

Theorem 3.6: Congruent Supplements Theorem

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

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43

Theorem 3.7: Congruent Complements Theorem

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

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44

Theorem 3.8: Vertical Angles Theorem

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

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45

Theorem 3.9

if two lines are perpendicular, then they intersect to form four right angles.

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46

Theorem 3.10

all right angles are congruent

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47

Theorem 3.11

perpendicular lines form congruent adjacent angles

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48

Theorem 3.12

if two angles are congruent and supplementary, then each angle is a right angle

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Theorem 3.13

if two congruent angles form a linear pair, then they are right angles

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50

parallel lines

coplanar lines that do not intersect

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51

skew lines

lines that do not intersect and are not coplanar

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52

parallel planes

planes that do not intersect

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53

transversal

a line that intersects two or more coplanar lines at different points

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54

interior angles

angles that lie between two transversals that intersect the same line

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55

exterior angles

the four outer angles formed by two lines cut by a transversal

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56

consecutive interior angles

interior angles that lie on the same side of the transversal

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57

alternate interior angles

nonadjacent interior angles that lie on opposite sides of the transversal

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58

alternate exterior angles

nonadjacent exterior angles that lie on opposite sides of the transversal

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59

corresponding angles

angles that lie on the same side of the transversal and in corresponding positions

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60

Theorem 3.14: Corresponding Angles Theorem

if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

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61

Theorem 3.15: Alternate Interior Angles Theorem

if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent

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Theorem 3.16: Consecutive Interior Angles Theorem

if two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary

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63

Theorem 3.17: Alternate Exterior Angles Theorem

if two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent

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64

Theorem 3.18: Perpendicular Transversal Theorem

in a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

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65

slope

the ratio of the change in the y-coordinate (rise) to the corresponding x-coordinate (run) as you move from one point to another along a line

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slope criteria

outlines a method for proving the relationship between lines based on a comparison of the slopes of the lines

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67

Postulate 3.12: Slopes of Parallel Lines

the slopes of two non-vertical lines are identical if and only if they are parallel; all vertical lines are parallel to other vertical lines

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Postulate 3.12: Slopes of Perpendicular Lines

two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals; vertical and horizontal lines are parallel

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69

slope-intercept form

y=mx+b, where m is the slope and b is the y-intercept of the line.

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70

point slope form

y-y₁=m(x-x₁)

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71

equation of a horizontal line

y=b, where b is the y-intercept

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72

equation of a vertical line

x=a, where a is the x-intercept

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73

Theorem 3.19: Converse of Corresponding Angles Theorem

if two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel

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74

Postulate 3.13

if given a line and a point not on that line, then there exists exactly one line through the point that is parallel to the given line

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75

Theorem 3.20: Alternate Exterior Angles Converse

if two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel

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Theorem 3.21: Consecutive Interior Angles Converse

if two lines are cut by a transversal so that consecutive interior angles are congruent, then the lines are parallel

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77

Theorem 3.22: Alternate Interior Angles Converse

if two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel

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78

Theorem 3.23: Perpendicular Transversal Converse

if two lines in a plane are perpendicular to the same line, then the lines are paralllel

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79

Distance Between a Point and a Line

given line segment AB and point C not on the line, there are an infinite number of lines that pass through the point and intersect the line

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80

Distance Between a Point and a Line

the distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point

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81

Postulate 3.14: Perpendicular Postulate

if given a line and a point not on that line, then there exists one line through that point that is perpendicular to the given line

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82

Distance Between Parallel Lines

the distance between two parallel lines is the perpendicular distance between one of the lines and any point on the other line

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83

Two Lines Equidistant from a Third

in a plane, if two lines are each equidistant from a third line, then the two lines are parallel to each other.

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