Download Geometry Chapter 1 Notes: Foundations for Geometry

Geometry Chapter 1 Notes: Foundations for Geometry
http://goo.gl/B5nXT
These notes are NOT entirely complete – there are places that I am leaving for the students to
help me fill out as an example the first few days.
Ch 1-1 Understanding Points, Lines & Planes
Undefined terms
Cannot be defined using other figures
Point
·
·
·
·
·
Line
· Straight path
· Has no thickness
· Goes on forever
(it CAN’T be measured)
· A single lowercase letter OR 2 capital
letters on the line with a line drawn over
the top
· Line n
Plane
· Flat surface
· Has no thickness
· Extends forever
· A single capital CURSIVE letter OR 3
points on the plane that are noncollinear
·
plane M Or plane ABC
Collinear
·
points that lie on the same line
Coplanar
·
points that lie in the same planes
Segment
· AKA “line segment”
· The part of the line between 2 points
· CAN be measured
· Named with 2 capital letters with a
segment drawn above them
Endpoint
· The starting points of a ray
· Named by a single capital letter
after all it IS a point J
· C or D
Names a location
Has no size
Represented by a dot
A single capital letter
R is read as “point R”
Ray
·
Starts from an endpoint
·
Goes on forever in one direction
(it CAN’T be measured)
· Named with 2 capital letters with a
segment drawn above them
· the endpoint is typically named first
Opposite rays
·
Two rays that have a common
endpoint
·
Together they form a line
Postulate
·
·
·
AKA “axiom”
Accepted as true without proof
Describe geometric properties
The postulates in this book are numbered in order of appearance in the book. The numbers have no
other significance and therefore, writing the numbers when I ask for a specific postulate will NOT be
acceptable. The EOC will not have any clue what you are talking about of you say “postulate 5-3-1”
Some postulates have names. Those names ARE acceptable as they are recognized throughout the
mathematical community.
Postulate 1-1-1
· Through any 2 points there is exactly 1 line
Postulate 1-1-2
· Through any 3 noncollinear points there is exactly one plane
Postulate 1-1-3
·If 2 points lie on a plane, then the line connecting the points also lies on
the plane
Postulate 1-1-4
·If 2 lines intersect, they intersect in exactly one point
Postulate 1-1-5
·If 2 planes intersect they intersect in exactly one line.
Ch 1-2 Measuring and Constructing Segments
Coordinate
· The number associated with a point on a ruler
Postulate 1-2-1
The Ruler Postulate
· Points on a line can be put in 1-1 correspondence with the real
numbers
Distance
· |a-b| or |b-a|
Length
· The distance between two points
· AB is read “the length of segment AB”
Congruent segments
· Segments that have the same length
Construction
· To use a compass and straightedge
Between
· 3 points on the same line such that if AB + BC = AC, then B is between
A and C
Postulate 1-2-2
Segment Addition
Postulate
· if B is between A and C , then AB + BC = AC
Midpoint
· divides a segment into 2 equal parts
Bisect
· to divide into 2 equal parts
Segment bisector
· a segment, ray or line that intersects a segment at its midpoint
· it can meet at any angle measure
Ch 1- 3 Measuring and Constructing Angles
Angle
·
Formed by 2 rays with a common
endpoint
· Naming angles is complex – a
diagram will be helpful!
Vertex
·
The common endpoint in an angle
Interior of an angle
·
The set of all points inside the sides
on an angle
Exterior of an angle
·
The set of all points outside an
angle
Measure
·
·
We will measure angles in degrees
Next year you will learn about
radians
Degree
·
There are 360 degrees in a circle
Postulate 1-3-1
The Protractor
Postulate
·
Acute angle
·
Measures BETWEEN 0◦ & 90◦
Right angle
·
Measures EXACTLY 90◦ degrees
Obtuse angle
·
Measures BETWEEN 90◦ and 180◦
Given a line AB, and O on the line,
all rays drawn from O can be put in
a 1-1 correspondence with the real
numbers from 0 to 180.
· In other words, a straight line
measure 180n degrees
Straight angle
·
Measures EXACTLY 180◦
Congruent angles
·
Angles that have the same measure
Postulate 1-3-2
Angle Addition
Postulate
·
If S is in the interior of angle ABC,
then angle ABS + angle SBC =
angle ABC
Angle bisector
·
A ray that divides an angle into 2
congruent angles
Ch 1-4 Pairs of Angles
Adjacent angles
2 angles in the same plane
Same vertex
Share a side
Do NOT share interior points
Linear Pair
A pair of adjacent angles whose
noncommon sides are opposite rays
Complementary
angles
2 angles with a sum of 90◦
May or may not be adjacent
Supplementary
angles
2 angles with a sum of 180◦
May or may not be adjacent
Vertical angles
2 nonadjacent angles formed by
intersecting lines
Think of a RailRoad sign
Ch 1 – 5 Using Formulas in Geometry
Perimeter
Sum of the side lengths
Area
Number of nonoverlapping square units that can cover a 2-D figure
Perimeter & Area of a
Rectangle
P = 2l + 2w
P = 2(l+w)
A = lw
Perimeter & Area of a
Square
P = 4s
A = s2
Perimeter & Area of a
Triangle
P=a+b+c
A = ½ bh
b is the length of the base
h is the height from the vertex
opposite the base meeting the
base at a 90 angle
Diameter
segment passing through the
center of a circle whose
endpoints are on the circle
Radius
segment whose endpoints are
the center of a circle and on the
edge of the circle
Circumference
the distance around the circle
Circumference and
Area of a circle
C = 2πr
C = πd
A = πr2
Ch 1- 6 Midpoint and Distance in the Coordinate Plane
Coordinate plane
- The x and y axes for
graphing
- 4 quadrants
- (x, y) coordinates are in
alphabetical order
Midpoint formula
-Used to find the middle of a
line segment
-The average of the x’s and
the average of the y’s
Distance formula
- Based upon the
Pythagorean theorem
·D=
Legs
The sides of a right triangle that make up the right angle
Hypotenuse
The longest side of a right triangle
Pythagorean theorem
The sum of the squares of the legs is equal to the square of
the hypotenuse
Ch 1- 7 Transformations in the Coordinate Plane
Transformation
A change in the position, size or
shape of a figure
Pre-image
Original figure
Image
-Resulting figure
-Named with prime notation
Reflection
-AKA “flip”
-A transformation across a line of
reflection
-Each point of the image and preimage are the same distance from
the line of reflection
Rotation
-AKA “turn”
-A transformation about a point,
called the point of rotation
-Each point in the pre-image and
image are the same distance from
the point of rotation
Translation
-AKA “slide”
-A transformation where all the
points in the pre-image move the
same distance in the same
direction