factorization in fields, intro to geometry

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

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

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

on the same line

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

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

have a common endpoint and form a straight line

1. a line contains at least two points

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

one and only one

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

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

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)

points that are all on one plane

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

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

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

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

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

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

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

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)

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

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

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.

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

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

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

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

an angle has one and only one bisector

theorems derived from theorems

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

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

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

if a=b and c≠0, then a/c = b/c

any number is equal to itself, or a=a

if a=b, then b=a

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

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

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

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

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

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

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

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

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 angles are congruent