Lecture 15: Angle Measure
15.1 Angle measure
Denition Suppose cfw_P, L, d is a Pasch geometry and A is the set of all angles in this
geometry. An angle measure, or protractor, is a function m : A (0, 180) such that (1)
if BC lies on the edge of a half pla
Lecture 10: Plane Separation
10.1 Convex sets
Denition We say a set of points S in a metric geometry is convex if for every two
distinct points P, Q S, P Q S.
Example
The set
S = cfw_(x, y) : (x, y) R2 , x2 + y 2 1
is convex set in the Euclidean Plane. Fo
Lecture 6: The Cartesian Plane Revisited
6.1 Points as vectors
If A = (x1 , y1 ), B = (x2 , y2 ), and R, then
Denition
A + B = (x1 + x2 , y1 + y2 ),
A = (x1 , x2 ),
A B = A + (1)B = (x1 x2 , y1 y2 ),
A, B = x1 x2 + y1 y2 ,
and
A, A .
A =
Given any A, B R2
Lecture 9: Angles and Triangles
9.1 Angles
Denition
the set
Given distinct noncollinear points A, B, and C in a metric geometry, we call
BA BC
an angle, which we denote ABC.
Note: An angle is a set of points, not a number. We will introduce the measure of
Lecture 5: Coordinate Systems
5.1 Special rulers
Theorem If f is a ruler for a line in a metric geometry cfw_P, L, d, then (1) g : R
dened by g(P ) = f (P ) and (2) h : R dened by h(P ) = f (P ) a, where a R,
are rulers for .
Proof If t R, then there exis
Lecture 4: Metric Geometry
4.1 Distance
Denition A distance function on a set S is a function d : S S R such that for all
P, Q S, (1) d(P, Q) 0, (2) d(P, Q) = 0 if and only if P = Q, and (3) d(P, Q) = d(Q, P ).
Example
Dene dE : R2 R2 R as follows: if P =
Lecture 8: Line Segments and Rays
8.1 Line Segments
Denition
Given distinct points A and B in a metric geometry, we call the set
AB = cfw_C : C AB and C = A, C = B, or A C B
the line segment from A to B.
Example Consider points A = (x1 , y1 ) and B = (x2
Lecture 16: The Moulton Plane
16.1 The Moulton plane
Example
Given b, m R, m > 0, let
Mm,b = cfw_(x, y) : (x, y) R2 , y = mx + b for x 0, y =
m
x + b for x > 0.
2
Let
LM = cfw_La : La LE , a R cfw_Lm,b : Lm,b LE , m 0, b R cfw_Mm,b : m > 0, b R.
Given dis
Lecture 12: Paschs Postulate
12.1 Paschs Postulate
Paschs Theorem Suppose cfw_P, L, d is a metric geometry which satises the plane
separation axiom. If is a line, ABC is a triangle, D , and A D B, then either
AC = or BC = .
In other words, in a metric ge
Lecture 7: Betweenness
7.1 Betweenness
Denition
If A, B, C are distinct, collinear points in a metric geometry cfw_P, L, d with
d(A, B) + d(B, C) = d(A, C),
then we say B is between A and C.
Notation: We write A B C to denote that B is between A and C. Mo
Lecture 14: Convex Quadrilaterals
14.1 Convex quadrilaterals
Denition Suppose cfw_A, B, C, D are four distinct points in a metric geometry, no three
of which are collinear, and int(AB), int(BC), int(CD), and int(DA) are disjoint. We call
ABCD = AB BC CD D
Lecture 13: Interiors
13.1 Interiors
Denition
If A and B are distinct points in a metric geometry, we call
int(AB) = AB cfw_A
the interior of the ray AB and we call
int(AB) = AB cfw_A, B
the interior of the segment AB.
Theorem
If A and B are distinct poin
Lecture 3: Incidence Geometry
3.1 Abstract geometry
Denition Suppose P is a set and L is a set of non-empty subsets of P such that (1)
for every A P and B P there exists L such that A and B , and (2) for
every L, has at least two elements from P. We call