Definition of Geometry
The branch of mathematics concerned
with properties of and relations
between points, lines, planes and
figures.
Origins Of Geometry
The earliest records of Geometry can be
traced to ancient Egypt and the Indus
Valley from around 3000 BC. Early
Geometry was a collection of observed
principles concerning lengths, angles,
area, and volumes. These principles were
developed to meet practical needs in
construction, astronomy, and other needs.
Origins of Geometry
Euclid, a Greek mathematician, wrote The
Elements of Geometry. It is considered
one of the most important early texts on
Geometry.
He presented geometry in a practical form
known as Euclidean geometry. Euclid was
not the first elementary Geometry
textbook, but the others fell into disuse
and were lost.
Chapter 1
Basics of Geometry
1.1 Patterns and Inductive
Reasoning
Vocabulary:
A conjecture is an unproven statement
that is based on observations.
Example:
3+4+5=4x3 6+7+8=7x3
4+5+6=5x3 7+8+9=8x3
5 + 6 + 7 = 6 x 3 8 + 9 + 10 = 9 x 3
Conjecture: The sum of any three
consecutive integers is 3 times the
middle number.
Example:
Conjecture: The sum of any two odd
numbers is ____.
1+1=2 1+3=4 3+5=8
7 + 11 = 18
13 + 19 = 32
Inductive reasoning is a process that
includes looking for patterns and
making conjectures. Prediction or
conclusion based on observation of a
pattern.
Example—expecting a traffic light to
stay green for 40 seconds because
you have seen it stay green for 40
seconds many times before.
Example—When you see the numbers
2, 4, 6, 8, and 10, and you expect
the next number to be 12.
Predict the next number.
2, 6, 18, 54, …
A counterexample is an example that
shows a conjecture is false.
Example: Show the conjecture is false
by finding a counterexample:
The difference of two positive
numbers is always positive.
Answer: 3 – 9 = -6
Describe pattern and predict next
number

1. 128, 64, 32, 16, …

2. 5, 4, 2, -1, …
Example

Given the pattern __, -6, 12, __, 48,
… answer the following exercises:
a. Fill in the missing numbers.
b. Determine the next two numbers in
this sequence.
c. Describe how you determined what
numbers completed the sequence.
d. Are there any other numbers that
would complete this sequence?
Answer to previous question.



A) 3, -6, 12, -24, 48
B) the next two numbers are -96 and
192
D) -24, -6, 12, 30, 48
next two numbers: 66 and 84
Complete the Conjecture
The sum of the first n even positive
integers is _____?
1st even integer:
2=
1(2)
Sum of 1st two even
pos. integers:
2+4=6=
2(3)
Sum of 1st three…:
2 + 4 + 6 = 12=3(4)
Sum of 1st four…: 2 + 4 + 6 + 8 = 20= 4(5)
Sum of 1st n even pos. Int. is: n(n + 1)

Counterexample


Show the conjecture is false by
finding a counterexample.
Conjecture: If the difference of two
numbers is odd, then the greater of
the two numbers must also be odd.
Finding the nth term

http://www.mathmagic.com/sequences/nth_term.htm
1.2 Points, Lines, Planes

The three undefined terms are point, line,
and plane.

A point has no dimension.

A point is usually named by a capital
letter.
‫٭‬A
All geometric figures consist of points.
Lines


Lines extend indefinitely and have no
thickness or width.
Lines are usually named by lower case
script letters or by writing capital letters
for 2 points on the line, with a double
arrow over the pair of letters.
l
line l
A
B
C
AB
BC
AC
Plane




A plane extends in two dimensions.
It is usually represented by a shape
that looks like a tabletop or wall.
A plane is a flat surface that extends
indefinitely in all directions.
A plane can be named by a capital
script letter or by three noncollinear
points in the plane. plane ABC or
plane R
http://www.mathopenref.com/plane.
html



Space-Three dimensional set of all
points.
Collinear points are points that lie on
the same line.
Coplanar points are points that lie on
the same plane.
A
Line Segment, Endpoint

A line segment is part of a line that
consists of two points, called
endpoints, and all points on the line
between the endpoints. Name by
using endpoints.
B

A
AB
Ray, Initial point
A ray is part of a line that consists of
a point, called an initial point, and all
points on the line that extend in one
direction. Name by using the
endpoint first, then any point of ray.

A
B
AB
Opposite Rays


If C is between A and B on AB, then
CA and CB are opposite rays.
B
C
A

Two or more geometric figures
intersect if they have one or more
points in common.
The intersection of two or more
geometric figures is the set of points
that the figures have in common.
Homework


Pages 6 – 9 #’s 12 – 15, 16 – 22
evens, 25 30, 47, 48
Pages 13 – 16 #’s 10 – 16 evens, 25
– 31 odds, 37, 44 - 47
1.3 Segments and their Measures
Postulates are rules that are
accepted without proof. Postulates
are also called axioms.
Ex: A line contains at least two points.
 A coordinate is a real number that
corresponds to a point on a line.
 The distance between two points on
a line is the absolute value of the
difference between the coordinates
of the points.





The length of a segment is the
distance between the endpoints.
When three points lie on a line, you
can say that one of them is between
the other two.
The Distance Formula is a formula
for finding the distance between two
points in a coordinate plane.
Congruent segments are segments
that have the same length.
Postulate 1: Ruler Postulate


The points on a line can be matched
one to one with real numbers. The
real number that corresponds to a
point is the coordinate of the point.
The distance between points A and
B, written AB, is the absolute value
of the difference between the
coordinates of A and B. AB is also
called the length of AB.
Finding distance between two
points

AB =







Find the distance between the points
12) E and A
13) F and B
14) E and D
15) C and B
16) F and A
Distance Formula: Given the two
points (x1, y1) and (x2, y2), the
distance between these points is
given by the formula:
( x 2  x1)  ( y 2  y1)
2
d=
2
Example
B (3, 4)
A(-1,2)
D(-3,-2)
C(4, -3)
Find AB, BC, CD, and AD.
Postulate 2: Segment Addition
Postulate


If B is between A and C, then
AB + BC = AC.
If AB + BC = AC, then B is between
A and C.



A lies between C and T. Find CT if
CA is 5 and AT is 8.
Find AC if CT is 20 and AT is 8.
See Notetaking guide book.
1.4 Angles and Their Measures

An angle consists of two noncollinear
rays that have the same initial point.
A
Name the angle:
<ABC or
<CBA or
BA and BC
B
1
<B or
<1
< ABC with sides
C
The initial point of the
rays is the vertex of the
angle. The vertex is
point B.
Congruent angles

Congruent angles are angles that
have the same measure.
m<A = m<B
A
B
Measure of an angle
In <AOB, ray OA and ray OB can be
matched one to one with the real
numbers from 0 to 180.
 The measure of <AOB is equal to the
absolute value of the difference
between the real numbers for
ray OA and ray OB.

Acute Angle

An acute angle is an angle that
measures between 0° and 90°.
http://optics.org/cws/article/research/23663
Angles

An angle separates
a plane into three
parts: the interior,
the exterior, and
the angle itself.
exterior
interior
Right Angle

A right angle is an angle that
measures 90°.
Obtuse Angle

An obtuse angle is an angle that
measures between 90° and 180°.
Straight Angle

A straight angle is an angle that
measures 180º.
Adjacent Angles

Two angles are adjacent if they have
a common vertex and side, but have
no common interior points.
D
A
A
B
<ABD AND <DBC
are adjacent
C
D
B
C
<ABD AND <DBC
are adjacent
Adjacent Angles
1
4
2
3
Adjacent Angles: <1 and <2; <2 and <3;
<3 and <4; <4 and <1
NOTE: Not adjacent
<1 and <3, <4 and <2
Name the angles in the figure
K
J
L
M
Protractor Postulate

For every angle there is a unique real
number r, called its degree measure,
such that 0 < r < 180.
*ILLUSTRATION ON NEXT SLIDE
A
BA
O
Angle Addition Postulate
If P is in the
interior of <RST,
then
m<RSP + m<PST =
m<RST
R

P
S
T
Measure the angle.
Then classify the
angle as acute,
right, obtuse, or
straight.
a. <AFD
b. <AFE
c. <BFD
d. <BFC
Homework
Pages 21 – 24 #’s 20, 24 – 30 evens,
31, 33, 35, 48, 55, 56
Pages 29 – 32 #’s 14 – 34 evens, 68,
70 - 73
1.5 Segment and Angle Bisector


A midpoint is the point that divides,
or bisects, a segment into two
congruent segments.
The midpoint M of PQ is the point
between P and Q such that PM = MQ.
P
M
Q
Midpoint of Segment

If B is the midpoint of segment AC
and AB = 2x + 8 and BC = 4x – 2
find AB, BC, and AC.
Construction




You can use a compass and a
straightedge to find the midpoint of a
segment.
A construction is a geometric
drawing that is created using a
limited set of tools, usually a
compass and a straightedge.
A compass is a tool used to draw
arcs.
A straightedge used to draw
segments.

Show a construction of a midpoint.
Midpoint Problem


Example:
If the coordinate of H is -5 and the
coordinate of J is 4, what is the
coordinate of the midpoint of HJ?
Midpoint Formula for Coordinate
Plane

Find the coordinates of the midpoint of AB with endpoints
A(-3, -4) and B(5, 5)
Midpoint Problem


The midpoint of JK is M(1, 4). One
endpoint is J(-3, 2). Find the
coordinates of the other endpoint.
The midpoint of PQ is M(5, 3). One
endpoint is P(-5, 12). Find the
coordinates of the other endpoint.
Bisect a segment

To bisect a segment or an angle
means to divide it into two congruent
parts.
Line l bisects segment
AB.
A
B
l
Segment bisector

Construct a segment bisector using a
compass and protractor.
Angle Bisector


An angle bisector is a ray that
divides an angle into two adjacent
angles that are congruent.
m<ABC = m<CBD
A
C
B
D
Bisect an angle

RT is the angle bisector of <QRS.
Given that m<QRS = 42º, what are
the measures of <QRT and <TRS?
Q
T
R
S
KM bisects <JKL.



m<JKM = 2x + 7
m<MKL = 4x – 41
Find m<JKM and m<MKL.
J
M
K
L
Angle Bisector

BD is the angle bisector of <ABC.
Find the m<ABD and m<DBC.
D
A
6x + 15
10x - 25
C
B
1.6 Angle Pair Relationships


Vertical angles consist of two angles
whose sides form two pairs of
opposite rays. Vertical angles are
congruent.
<1 and <3 and <2 and <4 are
vertical angles.
2
1
4
3
Linear Pair
A linear pair consists of two adjacent
angles whose noncommon sides are
opposite rays. Linear pairs of angles
are supplementary. (Sum of angles
is 180°.)
 <1 and <2, <2 and <3, <3 and <4,
and <4 and <1 are linear pairs of
angles.

2
1
4
3
Example




A) Are <1 and <2 a linear pair?
B) Are <4 and <5 a linear pair?
C) Are <5 and <3 vertical angles?
D) Are <1 and <3 vertical angles?
2
1
5
3
4
Example

Solve for x and y. Then find the
angle measures.
4x + 15
5x + 30
3y + 15
3y - 15
Example


A) Name one pair of vertical angles
and one pair of angles that form a
linear pair.
B) What is the measure of <GHI in
the figure above?
J
I
H
G
5x + 30
2x - 4
K
Complementary Angles


Complementary angles are two
angles whose measures have the
sum 90°.
Complement: The sum of the
measures of an angle and its
complement is 90°.
20
70
or
Supplementary Angles


Supplementary angles are two
angles whose measures have the
sum 180°.
Supplement: The sum of the
measures of an angle and its
supplement is 180°.
120
or
60
Examples




1. Given that <A is a complement of
<C and m<A = 47°, find m<C.
2. Given that <P is a supplement of
<R and m<R = 36°, find m<P.
3. <W and <Z are complementary.
The measure of <Z is 5 times the
measure of <W. Find m<W.
4. <T and <S are supplementary.
The measure of <T is half the
measure of <S. Find m<S.
Examples

5. When two lines intersect, the
measure of one of the angles they
form is 20° less than three times the
measure of one of the other angles
formed. What are the measures of
all four angles formed by the lines?
Homework
Pages 38 – 40 #’s 18, 24 – 30 evens,
38 – 42 evens, 44 - 49
Pages 47 – 50 #’s 8 – 26 evens, 28 –
36 evens, 43, 44, 46 – 52 evens
Combinations and Permutations



The rest of the slides in this chapter
come from these two sites. GLE’s
http://www.mathsisfun.com/combina
torics/combinationspermutations.html
http://www.glencoe.com/sec/math/p
realg/prealg03/extra_examples/chap
ter12/lesson12_7.pdf
Chapter 1 Test

Pages 60 – 63 Chapter Review and
Chapter Test
Combinations and Permutations



What’s the Difference?
Combination—order doesn’t matter.
Example: My fruit salad is a
combination of apples, grapes and
bananas. We don’t care what order
the fruits are in.


Permutation—order does matter. To
help you to remember, think
“Permutation …Position”
Example: The combination to the
safe was 472. We do care about the
order. “724” would not work, nor
would “247”. It has to be exactly
4-7-2.
Permutations

There are two types of permutations:
1. Repetition is Allowed: such as
the lock on previous slide. It
could be “333”.
2. No Repetition: for example the
first three people in a running
race. You can’t be 1st and 2nd.
Permutations with Repetition

Calculate-- If you have n things to choose
from, and you choose r of them, then the
permutations are:
r
n x n x … (r times) = n
where n is the number of things to choose
from, and you choose r of them
(Repetition allowed, order matters)
Example (Permutations with
repetition)

Combination Lock:
There are 10 numbers to choose from
(0, 1, …, 9) and you choose 3 of
them:
10 x 10 x … (3 times) = 10 =
1000
permutations
Ex: Perm. With Repetition


Police use photographs of various
facial features to help witnesses
identify suspects. One basic
information kit contains 195
hairlines, 99 eyes and eyebrows, 89
noses, 105 mouths, and 74 chins and
cheeks.
How many different faces can be
produced?
Answer

Number of faces:
195 x 99 x 89 x 105 x 74 =
Hairlines x eyes x noses x mouths x
chins
13,349,986,650
Example

The standard configuration for a New
York license plate is 3 digits followed
by 3 letters. How many different
license plates are possible if digits
and letters can be repeated?
New York
234 ABC
Answer


There are 10 choices for each digit
and 26 choices for each letter.
Number of plates:
10 x 10 x 10 x 26 x 26 x 26 =
17,576,000
Permutations without Repetition

In this case, you have to reduce the
number of available choices each
time.
Example w/o Repetition
For example, what order could 16
pool balls be in?
(After choosing, say, number “14”
you can’t choose it again.
1st choice: 16 possibilities,
2nd choice: 15 poss.,
Then 14, 13, etc.
16 x 15 x 14 x …=20,922,789,888,000

continued



Maybe you don’t want to choose
them all, just 3 of them, so that
would be only:
16 x 15 x 14 = 3360
In other words, there are 3,360
different ways that 3 pool balls could
be selected out of 16 balls.
Factorial Function

We can use the factorial symbol
( ! ) to help us write these functions
Mathematically.
4! = 4 x 3 x 2 x 1 = 24
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 =
5040


Select all billiard balls:
16! = 20,922,789,888,000
But if you wanted to select just 3:
16!
= 16 x 15 x 14
(16 – 3)!
= 3360
Formula
n

P(n, r) = Pr =
n
Pr
=
n!
(n – r)!
Where n is the number of things
choose from, and you choose r of
them (No repetition, order matters)
Example: Perm.
No repetition, Order matters


How many ways can 1st and 2nd place
be awarded to 10 people?
10!
(10 – 2)!
= 90
Example: Perm. no repetition



The standard configuration for a New
York license plate is 3 digits followed
by 3 letters.
How many different license plates
are possible if digits and letters
cannot be repeated?
10!
X
26! = 11,232,000
(10 – 3)!
(26 – 3)!
Example: Permutations

Twelve skiers are competing in the
final round of the Olympic freestyle
skiing aerial competition.
a) In how many different ways can
the skiers finish the competition?
(Assume there are no ties.)
b) In how many different ways can 3
of the skiers finish 1st, 2nd, and 3rd to
win the gold, silver, and bronze
metals?
Answers



A) There are 12! Different ways that
the skiers can finish the competition.
= 479,001,600
B) Any of the 12 skiers can finish 1st,
then any of the remaining 11 skiers
can finish 2nd, and finally any of the
remaining 10 skiers can finish 3rd.
12!
= 1320
(12 – 3)!
Perm. problem

You are considering 10 different
colleges. Before you decide to apply
to the colleges, you want to visit
some or all of them. In how many
orders can you visit (a) 6 of the
colleges and (b) all 10 colleges?
Answer

A)

B)
10
P
6
P
10
= 10!/(10 – 6)! = 151,200
=10!/(10 – 10)! =3,628,800
10
Note: 0! = 1
Permutation Example

How many ways can gold, silver, and
bronze medals be awarded for a race
run by 8 people?
Answer

P(8, 3) =
8!
=8x7x6
(8 – 3)! (8 choices for gold,
7 choices for silver,
6 choices for bronze)
There are 336 possible ways to award
the medals.
Example: Permutation

How many five-digit zip codes can be
made where all digits are unique?
The possible digits are the numbers
0 – 9.
answer
P(10, 5) = 10 x 9 x 8 x 7 x 6
10 choices for 1st digit
9 choices for 2nd digit
8 choices for 3rd digit
7 choices for 4th digit
6 choices for 3rd digit
= 30,240

Perm. problem


A classroom has 24 seats and 24
students. Assuming the seats are
not moved, how many different
seating arrangements are possible?
6.20 x 10^
23
COMBINATIONS
There are two types of
combinations (remember the order
does not matter now):
1. Repetition is Allowed: such as
coins in your pocket (5, 5, 5, 10,
10)
2. No Repetition: such as lottery
numbers (2, 14, 15, 27, 30, 33)

Combinations without Repetition


This is how the lotteries work. The
numbers drawn one at a time, and if
you have the lucky numbers (no
matter what order) you win!
Order in not important.
Combinations

Order does
matter
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
6 diff combinations
Order doesn’t
matter
1 2 3
One combination
Combination Formula
n


C(n, r) = C
=
r
=
C =
n
n!
r!(n – r)!
r
n
r
Example

A standard deck of 52 playing cards
has 4 suits with 13 different cards in
each suit as shown. If the order in
which the cards are dealt is not
important, how many different 5card hands are possible?
Answer


The number of ways to choose 5
cards from a deck of 52 cards is:
C
52
5
=
52!
5!(52 – 5)!
= 2,598,960
Examples

Your English teacher has asked you
to select 3 novels from a list of 10 to
read as an independent project. In
how many ways can you choose
which books to read?
Answer
10
C
3
10!
3!(10 – 3)!
= 120
Comb. problem


Your friend is having a party and has
15 games to choose from. There is
enough time to play 4 games. In
how many ways can you choose
which games to play?
1365
Combination Problem

How many ways can two slices of
pizza be chosen from a plate
containing one slice each of
pepperoni, sausage, mushroom, and
cheese pizza.
Answer

4
C
2
4!
2!(4 – 2)!
=6
Combination Problem

How many ways can three colors be
chosen from blue, red, green, and
yellow?
Answer

4
C
3
=
4!
3!(4 – 3)!
There are 4 ways to choose three colors from a list
of 4 colors.
Comb. Problem

Find the number of ways two
co-chairpeople can be selected for a
committee of 9 people.
Answer: 36


Angles in a polygon:
http://www.learner.org/channel/cour
ses/learningmath/measurement/sess
ion4/part_b/sum.html