factorization in fields, intro to geometry

hierarchy of geometry

hierarchy of geometry

undefined terms → definitions → postulates/axioms → theorems → corollaries

undefined terms

point, line, and plane (accepted as intuitive ideas; not defined)

line (A and h)

a straight, continuous arrangement of infinitely many points that extends forever in two directions

A is in/on h, h contains A, h passes through A

collinear

on the same line

line segment

two points (called endpoints) and all the points between them that are collinear with the two points

ray

a point on a line (called an endpoint) and all the points of the line that lie on one side of this point

opposite rays

have a common endpoint and form a straight line

two line postulates

1. a line contains at least two points

2. through any two points there exists one and only one line

existence and uniqueness

one and only one

ruler postulate

there is a unique measure that represents the difference between two endpoints

line intersection theorem (m, h, and O)

if two lines intersect, then they intersect at exactly one point

m and h intersect in/at O

O is the intersection of m and h

plane

a plane is a flat surface with no depth or boundary; can be named using three or more points on the plane or with a letter (capitalized)

coplanar points

points that are all on one plane

five postulates relating to planes

1. a plane contains at least three points not all on one line

2. through any three points, there is at least one plane, and through any three non-collinear points, there is exactly one plane

3. if two points are on a plane, then the line that contains the points is on that plane

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

5. a line can exist on an infinite number of planes

describing plane intersection (M, N, and [line] XY)

M and N intersect in [line] XY; [line] XY is the intersection of M and N; [line] XY is in M and N; M and N contain [line] XY

two theorems about planes

1. through a line and a point not in the line there is exactly one plane

2. if two lines intersect, then they lie exactly on one plane

betweenness

a point which lies between two other points on a line segment or a line (points are assumed to be collinear)

segment addition postulate

if C is between A and B, then AC + CB = AB

congruent objects

two objects that have the same size and shape are called congruent

definition of congruency

if two objects are congruent, then they have equal measures (bi-conditional)

definition of a midpoint

the midpoint of a segment is the point that divides the segments into two equal or congruent segments (not a bi-conditional; points have not been established as collinear)

definition of a bisector

if a line, ray, line segment, or plane intersects a segment at its midpoint, then it is the bisector of the segment

definition of an angle

an angle is formed by the union of two rays; the union is called its vertex

protractor postulate

informal:

for every angle, there is one and only one real number between 0 and 180 called the degree measure of the angle

formal:

suppose that H is a half-plane determined by [line] OA. then for any real number r such that 0 < r < 180, there is one and only one ray, [ray] OB such that B is in H and m<AOB = r.

definition of adjacent angles

two angles in a plane that have a common vertex, a common side, and no interior points in common are called adjacent angles

angle addition postulate

if B is in the interior of <AOC, then m<AOB + m<BOC = m<AOC

definition of congruent angles

two angles are said to be congruent if and only if they have the same measure. that is, <ABC ≅ <DEF if and only if m<ABC = m<DEF

definition of an angle bisector

a ray OC is a bisector of <AOB if and only if C is in the interior of the angle and <AOC ≅ <COB or m<AOC = m<COB

angle bisector postulate

an angle has one and only one bisector

corollaries

theorems derived from theorems

addition axiom

if a=b and c=d, then a+c=b+d

subtraction axiom

if a=b and c=d, then a-c=b-d

multiplication axiom

if a=b and c=d, then ac=bd

division axiom

if a=b and c≠0, then a/c = b/c

reflexive axiom

any number is equal to itself, or a=a

symmetric axiom

if a=b, then b=a

transitive axiom

if a=b and b=c, then a=c

substitution axiom

if a=b, then a can be replaced by b (and b by a)

properties of congruence

reflexive property, symmetric property, transitive property (not substitution property)

midpoint theorem

if M is the midpoint of [line segment] AB, then AM = ½AB and MB = ½AB

angle bisector theorem

if [ray] BX is the bisector of ∠ABC, then m​​∠ABX = ½m∠ABC and m∠XBC = ½m∠ABC

definition of supplementary angles

supplementary angles are two angles whose measure have the sum of 180. each angle is called the supplement of the other.

supplement axiom/postulate

if the exterior sides of two adjacent angles are opposite rays, then the angles are supplementary

definition of complementary angles

complementary angles are two angles whose measures have a sum of 90. each angle is called the complement of the other.

definition of vertical angles

vertical angles are two angles such that the sides of one angle are opposite rays to the sides of the other angle. if two lines intersect, then they form two pairs of vertical angles.

vertical angle theorem

vertical angles are congruent