1. Geometry
Chapter 1

2. Types of Geometry
Point
Line
Distance
Formula
Euclidean Synthetic Geometry
Exact location
Set of points extending in both directions, containing shortest paths between pts.
On a coordinatized (number) line, there is a ‘unique’ between points
|a - b| or
|b - a|
Euclidean Plane Coordinate Geometry
Ordered pair of real numbers
(x, y)
Set of ordered pairs (x, y) satisfying
Ax + By = C
On the coordinate plane, there is a ‘unique’ distance between two points
Pythag.
Theor.
Graph Theory
Node of a Network
Arc connecting nodes or connecting a node to itself
Distance not ‘unique’. There can be more than one line (arc) connecting two nodes.
Discrete Geometry
Dot
Set of dots in a row
Distance not ‘unique’. Two dots may not be part of the same line

3. Graph Theory - Nodes in a Network
Nodes are connected by one or more Arcs.
Odd nodes - have an odd number of connecting arcs
Even nodes - have an even number of connecting arcs
A network is traversable if does not have more than two odd nodes 2 or less).
A network is not traversable if it has more than two odd nodes (3 or more).

4. Undefined Terms (in this book)- point, line, plane
Definitions Using Undefined Terms:
Figure - a set of points
Space - the set of all points
Collinear - three or more points that are contained on the same line
Plane Figure - a set of points that are all in one plane
Coplanar - four or more points that are contained on the same plane

5. Point-Line-Plane Postulate (Euclidean Geometry)
a. Unique Line Assumption:
Through any two points there is exactly one line. If two points are in the same plane, the line containing them is in the plane.
b. Number Line Assumption:
Every line is a set of points (that can be put in a one-to-one correspondence with the real numbers), with one point on the line corresponding to 0 and another point corresponding to 1.
c. Dimension Assumption:
1. There are at least two points in space.
2. Given a line in a plane, there is at least one point on the plane that is not on the line.
3. Given a plane in space, there is at least one point in space that is not in the plane.

6. a. Unique Line AssumptionB A
b. Number Line Assumption 1
Point in plane, not on line
c. Dimension Assumption
Point in space, not on plane

7. Equation of a Line
Standard Form:
Oblique Line (not horizontal or vertical)
Ax + By = C ex. 4x + 3y = 7
(A, B, and C are integers, A is positive)
Horizontal Line
By = C ex. 3y = 9 --> y = 3
Vertical Line
Ax = C ex. 4x = 8 --> x = 2

8. Equation of a Line
Slope-Intercept Form:
Oblique Line (not horizontal or vertical)
y = mx + b ex. y = 5x - 2 m = slope
b = y-intercept (0, b)
Horizontal Line
y = b ex. y = 6 b = a real number
Vertical Line
x = c ex. x = -2 c = a real number

9. Slope of a Line Slope Formula: m = y2 - y1 x2 - x1
Oblique Line (not horizontal or vertical)
slope positive (line rises left to right): +/+ or -/-
slope negative (line falls left to right): +/- or -/+
Horizontal Line
Slope = 0 (‘y’ value remains constant, x varies)
Vertical Line
Slope - undefined (‘x’ value remains constant, y varies)

10. Standard Form --> Slope-Intercept Form Ax + By = C y = mx + b
Step 1: add or subtact ‘x’ term from both sides of the equation
ex. 4x + 3y = 9
-4x x
3y = -4x + 9

11. Standard Form --> Slope-Intercept Form Ax + By = C y = mx + b
Step 2: divide all terms by the coefficient of the ‘y’ term
ex. 3y = -4x + 9
y = -4/3x + 3
Slope = m = -4/3 y-intercept - (0,3)

12. Graphing an equation in Slope-Intercept Form
Step 1: Graph the y-intercept point (0, b) on the y-axis
Step 2: Use the slope to plot a second point
Step 3: Connect the points with a line (<---->)
Step 4: Check your slope (positive or negative)
a. Positive slopes - rise from left to right ( / )
b. Negative slopes - fall from left to right ( \ )

13. Give the equation of a line through two given points
Given: (x1, y1), (x2, y2) ex. (2, 1), (6, 4)
Step 1: Calculate the slope (using slope formula)
Step 2: Put slope into the slope-intercept equation form
y = mx + b ex. y = 3/4x + b
Step 3: Use one of the points to solve for ‘b’
y1 = m(x1) + b ex. 1 = 3/4(2) + b
1 = 3/2 + b
1 - 3/2 = b, b = -1/2
Step 4: Write equation y = 3/4x - 1/2

14. Solving a system of two equations
Given two equations (in standard form):
3x + 2y = 6 --> 3x + 2y = 6
x - y = 7 --> -3(x - y = 7) distribute
-3x + 3y = -21
Add first equation x + 2y = 6
Solve for one variable y = -15
y = -3
Plug ‘value’ into one of the original equations

15. Solving a system of two equations
Plug ‘value’ into one of the original equations and solve for the other variable.
x - y = 7 y = -3
x - (-3) = 7
x + 3 = 7
x = 7 - 3
x = 4 Solution (4, -3)
Check solution 3(4) + 2(-3) = 6
= 6 (true)

16. Intersecting, Parallel, and Perpendicular Lines
Intersecting Lines -
a. same line (same slope, same y-intercept)
b. different lines (different slopes)
Parallel Lines -
a. same slope, but different y-intercepts
Perpendicular Lines -
a. (slope of first line) * (slope of second line) = -1
the slopes are negative reciprocals (ex. 2/3 * -3/2 = -1)

17. One, Two, and Three Dimensional
One Dimensional corresponds to Collinear: (single axis)
ex. A number line, a taught string
Two Dimensional corresponds to Coplanar: (coordinate plane)
ex. The top of a table, the chalkboard
Three Dimensional (xy plane, yz plane, and xz plane combined)
ex. Ice cream cone, basketball, house

18. Triangle Inequality Postulate
(∆ABC) - sides AB, BC, and CA ex. 6, 5, 7
AB + BC > CA > 7 (true)
BC + CA > AB > 6 (true)
CA + AB > BC > 5 (true)
If one of the inequalities is not true, then the points will not form a triangle (ex. 6, 5, and 11 cannot be the lengths of the sides of a triangle).