Studied by 2 peopleconjecture
an educated guess based on known information and specific examples
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
statement
any sentence that is either true or false, but not both
truth value
the truth or falsity of a statement
compound statement
two or more statements joined by the word "and" or "or"
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 "^"
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"
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
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"
hypothesis
the phrase immediately following the word "if" in a conditional statement
conclusion
the phrase immediately following the word "then" in a conditional statement
converse
the statement formed by exchanging the hypothesis and conclusion of a conditional statement
inverse
the statement formed by negating both the hypothesis and conclusion of a conditional statement
contrapositive
the statement formed by negating both the hypothesis and conclusion of the converse of a conditional statement
biconditional statement
inductive reasoning
the process of reaching a conclusion based on a pattern of examples; assuming that an observed pattern may continue
deductive reasoning
using general facts, rules, definitions, or properties to reach specific valid conclusions from given statements
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
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
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)
Postulate 3.1 (the numbers of the postulates are not necessary to know)
through any two points, there is exactly one line
Postulate 3.2
through any three noncollinear points, there is exactly one plane
Postulate 3.3
a line contains at least two points
Postulate 3.4
a plane contains at least three noncollinear points
Postulate 3.5
if two points lie in a plane, then the entire line containing those points lies in that plane
Postulate 3.6
if two lines intersect, then their intersection is exactly one point
Postulate 3.7
if two planes intersect, then their intersection is a line.
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
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.
deductive argument
an argument that proves a statement by building a logical chain of statements and reasons
flow proof
a type of proof that uses boxes and arrows to show the flow of a logical argument
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
Postulate 3.8: Ruler Postulate
the points on any line or line segment can be put into one-to-one correspondence with real numbers.
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
Postulate 3.10: Protractor Postulate
every angle has a measure that is between 0 and 180
Postulate 3.11: Angle Addition Postulate
D is in the interior of <ABC if and only if m<ABD + m<DBC = m<ABC
Theorem 3.3: Supplement Theorem
if two angles form a linear pair, then they are supplementary angles
Theorem 3.4: Complement Theorem
if the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.
Theorem 3.5: Reflexive Property of Congruence
any geometric object is congruent to itself
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.
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.
Theorem 3.6: Congruent Supplements Theorem
angles supplementary to the same angle or to congruent angles are congruent
Theorem 3.7: Congruent Complements Theorem
angles complementary to the same angle or to congruent angles are congruent
Theorem 3.8: Vertical Angles Theorem
if two angles are vertical angles, then they are congruent.
Theorem 3.9
if two lines are perpendicular, then they intersect to form four right angles.
Theorem 3.10
all right angles are congruent
Theorem 3.11
perpendicular lines form congruent adjacent angles
Theorem 3.12
if two angles are congruent and supplementary, then each angle is a right angle
Theorem 3.13
if two congruent angles form a linear pair, then they are right angles
parallel lines
coplanar lines that do not intersect
skew lines
lines that do not intersect and are not coplanar
parallel planes
planes that do not intersect
transversal
a line that intersects two or more coplanar lines at different points
interior angles
angles that lie between two transversals that intersect the same line
exterior angles
the four outer angles formed by two lines cut by a transversal
consecutive interior angles
interior angles that lie on the same side of the transversal
alternate interior angles
nonadjacent interior angles that lie on opposite sides of the transversal
alternate exterior angles
nonadjacent exterior angles that lie on opposite sides of the transversal
corresponding angles
angles that lie on the same side of the transversal and in corresponding positions
Theorem 3.14: Corresponding Angles Theorem
if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.
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
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
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
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.
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
slope criteria
outlines a method for proving the relationship between lines based on a comparison of the slopes of the lines
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
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
slope-intercept form
y=mx+b, where m is the slope and b is the y-intercept of the line.
point slope form
y-y₁=m(x-x₁)
equation of a horizontal line
y=b, where b is the y-intercept
equation of a vertical line
x=a, where a is the x-intercept
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
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
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
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
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
Theorem 3.23: Perpendicular Transversal Converse
if two lines in a plane are perpendicular to the same line, then the lines are paralllel
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
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
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
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
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.
Note
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