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Transcript
Chapter 1
Basics of Geometry
Chapter Objectives









Using Inductive Reasoning to identify patterns.
Identify collinear characteristics.
Utilize Distance and Midpoint Formulas
Label acute, obtuse, right, and straight angles.
Identify angle measures
Apply Angle Addition Postulate
Compare Complimentary v Supplementary angles
Define angle and segment bisectors
Identify the basics of perimeter, circumference, and
area
Lesson 1.1
Patterns
And
Inductive Reasoning
Lesson 1.1 Objectives



Identify patterns in numbers and shape
sequences.
Use inductive reasoning.
Define Geometry.
Geometry is…


Geometry is a branch of mathematics that
deals with the measurement, properties, and
relationships of points, lines, angles, surfaces,
and solids; the study of properties of given
elements that remain invariant under
specified transformations.
Basically what that means is geometry is the
study of the laws that govern the patterns
and elements of mathematics.
Definition from Merriam-Webster Online Dictionary.
Inductive Reasoning



Inductive Reasoning is the process in
which one looks for patterns in samples and
makes conjectures of how the pattern will
work for the entire population.
A conjecture is an unproven statement
based on observations.
A conjecture is math’s version of a
hypothesis, or educated guess.

The education comes from the observation.
Using Inductive Reasoning

Much of the reasoning in Geometry
consists of three stages
1.
2.
3.
Look for a Pattern. Look at examples and
organize any ideas of a pattern into a diagram or
table.
Make a Conjecture. Use the examples to try
to identify what step was taken to get from
element to element in the pattern.
Verify the Conjecture. Use logical reasoning
to verify the conjecture is true for all cases.
Example 1
Identify the next member of the group:
1 , 4 , 7 , 10
13
1 , 4 , 9 , 16
25
Counterexamples


A counterexample is one example
that shows a conjecture is false.
Therefore to prove a conjecture is true,
it must be true for all cases.
Conjecture: Every month has at least 30 days.
Counterexample: February has 28 (or 29).
Goldbach’s Conjecture

In the early 1700s a Prussian mathematician
named Goldbach noticed that many even
numbers greater than 2 can be written as the
sum of two primes.
4=2+2
6=3+3
8=3+5
10 = 3 + 7
This conjecture is unproven
for all cases, but has been
proved for all even numbers
up to 4 x 1014.
400,000,000,000,000
Homework 1.1

In-class

1-11


Homework


Page 6-9
12-26 ev, 27, 28-46 ev, 47, 48, 52-70 even
Due Tomorrow
Lesson 1.2
Points, Lines, and Planes
Lesson 1.2 Objectives


Define the basic terms of geometry.
Sketch the basic components of
geometrical figures.
Start-Up

Give your definition
of the following



Point
Line
Plane

Not an airplane!


These terms are
actually said to be
undefined, or have
no formal definition.
However, it is
important to have a
general agreement
on what each word
means.
Point

A point has no dimension.



Meaning it takes up no space.
It is usually represented as a dot.
When labeling we designate a capital
letter as a name for that point.

We may call it Point A.
A
Line

A line extends in one dimension.




Meaning it goes straight in either a vertical, horizontal, or
slanted fashion.
It extends forever in two directions.
It is represented by a line with an arrow on each
end.
When labeling, we use lower-case letters to name the
line.


Or the line can be named using two points that are on the
line.
So we say Line n, or AB
n
B
A
A
Plane



B
A plane extends in two dimensions.


M
C
Meaning it stretches in a vertical direction as well as a
horizontal direction at the same time.
It also extends forever.
It is usually represented by a shape like a tabletop or
a wall.
When labeling we use a bold face capital letter to
name the plane.


Plane M
Or the plane can be named by picking three points in the
plane and saying Plane ABC.
Collinear
The prefix co- means the same, or to
share.
 Linear means line.
 So collinear means that points lie on
the same line.

A
B
C
We say that points A, B, and C are collinear.
Coplanar

Coplanar points are points that lie on
the same plane.
A
M
C
B
So points A, B, and C are said to be coplanar.
Line Segment

Consider the line AB.


It can be broken into smaller pieces by
merely chopping the arrows off.
This creates a line segment or
segment that consists of endpoints A
and B.

This is symbolized as AB
B
A
Ray

A ray consists of an initial point where
the figure begins and then continues in
one direction forever.


It looks like an arrow.
This is symbolized by writing its initial
point first and then naming any other
point on the ray, AB .

Or we can say ray AB.
B
A
Opposite Rays

If C is between A and B on a line, then
ray CA and ray CB are opposite rays.

Opposite rays are only opposite if they are
collinear.
A
C
B
Intersections of
Lines and Planes

Two or more geometric figures intersect if they
have one or more points in common.




If there is no point or points shown, they the figures do not
intersect.
The intersection of the figures is the set of points the
figures have in common.
Two lines intersect at one point.
Two planes intersect at one line.
A
m
n
Homework 1.2

In-class

1-8


Homework


Page 13-15
9-42 every 3rd, 44 – 51, 56-66 ev, 68-75
Due Tomorrow
Lesson 1.3
Segments and Their Measures
Lesson 1.3 Objectives




Define what a postulate is.
Use segment postulates.
Utilize the Distance Formula.
Identify congruent segments.
Definition of a Postulate

A postulate is a rule that is accepted
without a proof.



They may also be called an axiom.
Basically we do not need to know the
reason for the rule when it is a
postulate.
Postulates are used together to prove
other rules that we call theorems.
Postulate 1: Ruler Postulate


The points on a line can be matched to
real numbers called coordinates.
The distance between the points, say
A and B, is the absolute value of the
difference of the coordinates.

Distance always positive.
4
8
A
B
Length

Finding the distance between points A
and B is written as


AB
Writing AB is also called the length of
line segment AB.
Betweenness

When three points lie on a line, we can
say that one of them is between the
other two.


A
This is only true if all three points are
collinear.
We would say that B is between A and C.
B
C
Postulate 2: Segment Addition Postulate

If B is between A and C, then


AB + BC = AC.
Also, the opposite is true.

If AB + BC = AC, then B is between A and
C.
BC
AB
A
B
AC
C
Lesson 1.3A Homework

In-class

1, 3, 4-8 ev, 11


Homework



p21-22
13-33
Due Tomorrow
Quiz Monday

Lessons 1.1-1.3
Lesson 1.3
Part II
Segment Addition Postulate Review

Identify the unknown lengths given that
BD=4, AE=17, AD=7, and BC=CD
A
 BC


3
AC


2
AB


B
5
DE

10
C
D
E
Distance Formula
To find the distance on a graph between two
points
A(1,2)
AB =
B(7,10)
We use the Distance Formula
(x2 – x1)2 + (y2 – y1)2
Distance can also be found using the Segment Addition
Postulate, which simply adds up each segment of a line
to find the total length of the line.
Example 2

Using the Distance Formula, find the length of
segment OK with endpoints


O(2,6)
K(5,10)
(x2 – x1)2 + (y2 – y1)2
(5 – 2)2 + (10 – 6)2
32 + 42
9 + 16
25
=5
Example 3

This is one part of the problem for #34

Find the distance between points A and C.


A(-4,7)
C(3,-2)
(x2 – x1)2 + (y2 – y1)2
(3 – -4)2 + (-2 – 7)2
72 + (-9)2
49 + 81
130
Congruent Segments

Segments that have the same length
are called congruent segments.

This is symbolized by =.
Hint: If the symbols are there, the
congruent sign should be there.
LE = NT
If you want to state
two segments are
congruent, then you
write
LE = NT
If you want to state
two lengths are equal,
then you write
Lesson 1.3B Homework

In-Class

4-8 ev


Homework

37-56, 60-70 ev



p22-24
skip 44,47,52-54
Due Tomorrow
Quiz Tomorrow

Lesson 1.1-1.3
Lesson 1.4
Angles and Their Measures
Lesson 1.4 Objectives




Use the angle postulates.
Identify the proper name for angles.
Classify angles as right, obtuse, acute,
or straight.
Measure the size of an angle.
What is an Angle?



An angle consists of two different rays
that have the same initial point.
The rays form the sides of the angle.
The initial point is called the vertex of
the angle.

Vertex can often be thought of as a corner.
Naming an Angle

All angles are named by using three points


Name a point that lies on one side of the angle.
Name the vertex next.


The vertex is always named in the middle.
Name a point that lies on the opposite side of
the angle.
So we can call It
Or NOW
O
WON
W
N
Congruent Angles

Congruent angles are angles that have the
same measure.

To show that we are finding the measure of an
angle

Place a “m” before the name of the angle.
m WON =m NOW
Equal Measures
WON =
NOW
Congruent Angles
Types of Angles
Acute
Right
Obtuse
Straight
<90
=90
>90
=180
Looks
like
Measure
Other Parts of an Angle


The interior of an angle is defined as the set
of points that lie between the sides of the
angle.
The exterior of an angle is the set of points
that lie outside of the sides of the angle.
Exterior
Interior
Postulate 4: Angle Addition Postulate

The Angle Addition Postulate allows
us to add each smaller angle together
to find the measure of a larger angle.
What is the
total?
49o
32o
17o
Adjacent Angles

Two angles are adjacent angles if
they share a common vertex and side,
but have no common interior points.

Basically they should be touching, but not
overlapping.
C
CAT and TAR
are adjacent.
CAR and TAR
are not adjacent.
T
A
R
Homework Lesson 1.4

In-Class

1-16


Homework

18-48 ev, 50-53, 62-78 ev


p29-32
skip 30,32,34
Due Tomorrow
Lesson 1.5
Segments
and
Angle Bisectors
Lesson 1.5 Objectives



Identify a segment bisector
Identify an angle bisector
Utilize the Midpoint Formula
Congruence marks are used to show that
Midpoint

The midpoint of a segment is the point that divides
the segment into two congruent segments.


segments are congruent. If there is more
than one pair of congruent segments, then
each pair should get a different number of
congruence marks.
The midpoint bisects the segment, because bisect means to
divide into two equal parts.
A segment bisector is a segment, ray, line, or plane
that intersects the original segment at its midpoint.
H
J
We say that O is the midpoint
of line segment JY.
O
T
Now we can say
line HT is a
Y
segment bisector of segment JY.
Midpoint Formula
We can also find the midpoint of segment AB by
using its endpoints in…
The Midpoint Formula
A(1,2) B(7,10)
Midpoint of AB =
(
(y1 + y2)
(x1 + x2)
2
,
2
)
This gives the coordinates of the midpoint, or point that
is halfway between A and B.
Example 4

This is an example of how to determine the midpoint
knowing the two endpoints.
A(1,2)
B(7,10)
(
(
(
(
(x1 + x2)
2
(1 + 7)
2
8
2
4
,
,
,
,
(y1 + y2)
2
(2 + 10)
2
12
2
6
)
)
)
)
Example 5

This is an example of how to find an endpoint
knowing the midpoint and the other endpoint.

Say the midpoint is (8,5) and one endpoint is (4,9).
Remember that each
coordinate from the
midpoint was found from…
(x1 + x2)
2
= 8=
(4 + x2)
(
(x1 + x2)
2
,
(y1 + y2)
2
(y1 + y2)
x2
-4
-4
2
So the coordinates for
the other endpoint are
x2
= 12
(12,1)
x2
2
16 = 4 + x2
)
So, use each
coordinate from the
midpoint formula to
solve for x2.
=5=
(9 + y2)
x2
2
10 = 9 + x2
-9
-9
y2
=1
x2
Short Cut to Find Endpoint


Say the midpoint is (8,5) and one endpoint is (4,9).
Remember that the midpoint is half way between the
endpoints.
Add 4 to x
(4,9)
Minus 4 from y
(8,5)
Add 4 to x
(12,1)
Minus 4 from y
Angle Bisector

An angle bisector is a ray that divides
an angle into two adjacent angles that
are congruent.

To show that angles are congruent, we use
congruence arcs.
Homework 1.5

In-Class

1-2, 4-13


Homework


p38-42
18-32 ev, 38-54, 62-72 ev
Due Tomorrow
Lesson 1.6
Angle Pair Relationships
Lesson 1.6 Objectives



Identify vertical angle pairs.
Identify linear pairs.
Differentiate between complementary
and supplementary angles.
Vertical Angles

Two angles are vertical angles if their sides form
two pairs of opposite rays.


To identify the vertical angles, simply look straight
across the intersection to find the angle pair.


Basically the two lines that form the angles are straight.
Hint: The angle pairs do not have to be vertical in position.
Vertical Angle pairs are always congruent!
4
1
3
2
4
1
3
2
Linear Pair

Two adjacent angles form a linear pair if
their non-common sides are opposite rays.


Simply put, these are two angles that share a
straight line.
Since they share a straight line, their sum is…

180o
1
2
Complementary v Supplementary

Complementary
angles are two
angles whose sum is
90o.

Complementary
angles can be
adjacent or nonadjacent.

Supplementary
angles are two
angles whose sum is
180o.

Supplementary
angles can be
adjacent or nonadjacent.
Homework 1.6

In-Class

1, 4-7


Homework



p47-50
8-40 ev, 46-54 ev, 60-74 ev
Due Tomorrow
Test Tuesday
Lesson 1.7
Intro to Perimeter,
Circumference and
Area
Lesson 1.7 Objectives


Find the perimeter and area of common
plane figures.
Establish a plan for problem solving.
Perimeter and Area of a Rectangle


Recall that the
perimeter of a figure
is the sum of the
lengths of the sides.
A rectangle has two
pairs of opposite sides
that are congruent.
 l+w+l+w=

P = 2l + 2w


Recall the area of a
figure is the measure of
space inside the figure.
This is found by taking
the length of the
rectangle times the
width of the rectangle.
 l•w =
 A = lw
Perimeter and Area of a Square

Since a square has four
congruent sides, the
formula is quite
simple…

s+s+s+s=
 P = 4s
s

Since a square is also a
rectangle, we can find
the area by multiplying
the length times the
width

s•s=
2
 A = s
Perimeter and Area of a Triangle

The perimeter can be
found by adding the
three sides together.


P=a+b+
bc
If the third side is
unknown, use the
Pythagorean Theorem
c
to solve for the
unknown side.

a2 + b2 = c2

Where a,b are the two
shortest sides and c is
the longest side.

h
The area of a triangle is
half the length of the
base times the height
a
of the triangle.


The height of a triangle
is the perpendicular
length from the base to
the opposite vertex of
the triangle.
A = ½bh
Circumference and Area of a Circle


The perimeter of a
circle is called the
circumference.
It is found by taking the
diameter times .


The area is found by
taking pi times the
radius squared.

A = r2
C = d or 2r

r is the radius, which is
half the diameter.
d=diameter
r=radius
Homework

In-Class

1-8


Homework



p55-58
10-32 ev, 38-48 ev, 58-62 ev
Due Tomorrow
Test Tuesday

December 18th