Geometry Midterm/Module 3

conjecture

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.