Download Chapter 1 Notes.notebook

Chapter 1 Notes.notebook
September 12, 2012
1.1 Patterns and Inductive Reasoning
Inductive Reasoning:
Reasoning based on patterns you observe
What are the next 2 terms in each sequence?
1, 2, 4, 7, 11, Conjecture:
A conclusion you reach using inductive reasoning
Make a conjecture about the sum of the first 25 odd numbers.
1 = 1 =
1 + 3
= 4 =
1 + 3 + 5 =
1 + 3 + 5 + 7=
=
=
Conjecture: Chapter 1 Notes.notebook
September 12, 2012
Counterexample:
An example for which the conjecture is incorrect.
Can you find a counterexample for the following conjecture?
Conjecture
The square of any number is greater than the original number.
Conjecture
The sum of two numbers is greater than either number.
Chapter 1 Notes.notebook
September 12, 2012
1.3 Points, Lines and Planes
A point is a location, it has no size .
Space is defined as the set of all points.
A line is a series of points that extends in two opposite directions without end.
line a
Points that lie on the same line are collinear
A plane is a flat surface that has no thickness. A plane contains many lines and extends without end.
Points and lines in the same plane are coplanar.
Chapter 1 Notes.notebook
September 12, 2012
A postulate or axiom is an accepted statement of fact.
Postulate 1.1
Through any two points there is exactly one line.
Postulate 1.2
If two lines intersect, then they intersect in exactly one point.
Postulate 1.3
If two planes intersect, then they intersect in exactly one line.
Postulate 1.4
Through any 3 noncollinear points, there is exactly one plane.
p.6; 2­38 even
p.9, 54
p.19;2­42 even,50­60
Chapter 1 Notes.notebook
September 12, 2012
1.4 Segments, Rays, Parallel Lines and Planes
A segment is the part of a line consisting of two endpoints and all points between them.
A ray is the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint.
Opposite rays are two collinear rays with the same endpoint. Opposite rays always form a line.
Parallel Lines are coplanar lines that do not intersect.
Skew Lines are noncoplanar; they are not parallel and do not intersect.
Parallel Planes are planes that do not intersect. A line and a plane that do not intersect are also parallel.
Chapter 1 Notes.notebook
September 12, 2012
Solving Linear Equations (Review)
p.25;1­15,25­33
p.30;1­20
Chapter 1 Notes.notebook
September 12, 2012
Chapter 1 Notes.notebook
September 12, 2012
Measuring
1.5 Segments
AC = 52cm
Find x.
C
B
A
Segment Addition Postulate
If three points A, B and C are collinear and B is between A and C, then AB + BC = AC
A
C
B
same shape and
congruent
same size
A midpoint of a segment is a point that divides the segment into two congruent segments. A midpoint of a segment bisects the segment.
X
Y
Z
C is the midpoint of Find AC, CB and AB.
A
C
B
Chapter 1 Notes.notebook
September 12, 2012
1.6 Measuring Angles
An angle is formed by two rays with the same endpoint. The endpoint is called the vertex of the angle.
D
Naming Angles
Q
1
J
Classifying Angles
Two angles with the same measure are congruent.
≅ Congruent
1
the measure of angle 1 = the measure of angle 2
2
Chapter 1 Notes.notebook
September 12, 2012
Angle Addition Postulate
If point B is in the interior of AQC, then m AQB + m BQC = m AQC.
B
A
Q
C
m XQZ = 157 m XQY = 36
Z
Y
Find m YQZ Q
X
Angle Pairs
Vertical Angles
Two angles whose sides are opposite rays
(two intersecting lines form vertical angles)
Complementary Angles
Adjacent Angles
Two angles that have a common side and the same vertex.
Supplementary Angles
Two angles whose measures have sum
Two angles whose measures have sum Name a pair of:
L
Vertical Angles
M
Adjacent Angles
Q
P
Complementary
Angles
N
O
Supplementary
Angles
NQL
LQM
p.33; 8­19, 39­52
HW: p.40; 1­34; 42­46
PQL
OQP
Chapter 1 Notes.notebook
September 12, 2012
Chapter 1 Notes.notebook
September 12, 2012
To simplify radical expressions, remove perfect­square factors from the radical.
Rationalizing the Denominator
Chapter 1 Notes.notebook
Adding Like Radicals
September 12, 2012
Chapter 1 Notes.notebook
September 12, 2012
Simplest Radical Form
A radical is in simplest form when:
• The radicand has no perfect square factors
• The radicand has no fractions
• The denominator of a fraction has no radical
Chapter 1 Notes.notebook
September 12, 2012
1.8 The Coordinate Plane
Find the distance between any two points Distance formula:
Find the length of
A (0, ­2)
B (3, 3)
II
I
III
IV
Chapter 1 Notes.notebook
September 12, 2012
To find the midpoint of any segment with endpoints
Average
the x
and y
coordinates.
Midpoint formula
Find the midpoint of Y
X
M is the midpoint of Find the coordinates of point B.
A
M
Find MQ to the nearest tenth and the coordinates of the midpoint of Q
M
(14, ­2) (7, ­8)
Chapter 1 Notes.notebook
September 12, 2012
1.9 Perimeter, Circumference and Area
Rectangle
Square
Circle
s
s
s
s
Find:
•
•
•
•
Perimeter of the picture
Perimeter of the frame
Area of the picture
Area of the picture + frame
6" x 7" picture
1/2" frame
Find the circumference and area of the circle, in terms of π
13 ft
Chapter 1 Notes.notebook
Graph quadrilateral KLMN with vertices
K (­3, ­3), L (1, ­3), M (1, 4) N (­3, 1)
Find perimeter and area of KLMN
September 12, 2012
Chapter 1 Notes.notebook
September 12, 2012