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Geometry
Chapter 11
Informal Study of Shape
• Until about 600 B.C. geometry was pursued in response to practical,
artistic and religious needs. Considerable knowledge of geometry
was accumulated, but mathematics was not yet an organized and
independent discipline.
• Beginning in about 600 B.C. Pythagoras, Euclid, Thales, Zeno,
Eudoxus and others began organizing the knowledge accumulated
by experience and transformed geometry into a theoretical science.
• NOTE that the formality came only AFTER the informality of
experience in practical, artistic and religious settings!
• In this class, we return to learning by trusting our intuition and
experience. We will discover by exploring using picture
representations and physical models.
Informal Study of Shape
• Shape is an undefined term.
• New shapes are being discovered all the
time.
• FRACTALS
Informal Study of Shape
Our goals are:
• To recognize differences and similarities
among shapes
• To analyze the properties of a shape or
class of shapes
• To model, construct and draw shapes in a
variety of ways.
NCTM Standard
Geometry in Grades Pre-K-2
Children begin forming concepts of shape long before
formal schooling. They recognize shape by its
appearance through qualities such as “pointiness.” They
may think that a shape is a rectangle because it “looks
like a door.”
Young children begin describing objects by talking about
how they are the same or how they are different.
Teachers will then help them to gradually incorporate
conventional terminology. Children need many
examples and nonexamples to develop and refine their
understanding.
The goal is to lay the foundation for more formal geometry
in later grades.
•
• Point
A
• Line
<
• Collinear
C
B
<
• Plane
B
>
C
D
F
E
BC
G
>
• If two lines intersect, their intersection is a
point, called the point of intersection.
• Parallel Lines
<
<
>
>
• Concurrent
• Skew Lines – nonintersecting lines that are not
parallel.
H
• Line segment
• Endpoint
• Length
I
HI
M
• Congruent
K
J
L
JK  LM
m(JK) = m(LM)
JK  LM
• Midpoint
N
O
NO  OP
P
<
A
X
B
>
• Half Line
• A point separates a line into 3 disjoint sets:
The point, and 2 half lines.
• Ray - the union of a half line and the point.
S
T
> ST
• Angle – the union of two rays with a common
endpoint.
U
W
V UWV
VWU
W
• Vertex: W Common endpoint of the two rays.
• Sides:
WU and WV
Interior
Exterior
• The angle separates the plane into 3 disjoint
sets: The angle, the interior of the angle, and
the exterior of the angle.
Interior
Exterior
• Degrees
• Protractor
• Zero Angle: 0°
• Straight Angle: 180°
• Right Angle: 90°
>
<
>
>
^
>
• Acute Angle: between 0° and 90°
• Obtuse Angle: between 90° and 180°
• Reflex Angle
• Perpendicular Lines
Adjacent Angles
2
3
1
6
4
5
Adjacent Angles
2
1
3
6
4
5
Vertical Angles
2
3
1
6
4
5
Vertical Angles
2
1
6
3
4
5
1
2
The sum of the measures of Complementary
Angles is 90°.
• Complementary angles
60
30
• Adjacent complementary angles
1
2
The sum of the measures of
Supplementary Angles is 180°.
• Supplementary Angles
150
30
• Adjacent Supplementary Angles
• Lines cut by a Transversal – these lines are not
concurrent.
• Transversal
• Corresponding Angles
1
2
4
3
6
5
8
7
• Transversal
• Corresponding Angles
1
2
4
3
6
5
8
7
• Parallel lines Cut by a Transversal
• Parallel lines Cut by a Transversal
• Corresponding Angles
2
1
4
3
5
6
8
7
• Parallel lines Cut by a Transversal
• Corresponding Angles
2
1
4
3
5
6
8
7
• Describe the relative position of angles 3 and 5.
• What appears to be true about their measures?
2
1
4
3
5
6
8
7
• Alternate Interior Angles
2
1
4
3
5
6
8
7
• Describe the relative positions of angles 1 and 7.
• What appears to be true about their measures?
2
1
4
3
5
6
8
7
• Alternate Exterior Angles
1
2
4
3
5
6
8
7
Triangle
The sum of the measure of the interior angles of
any triangle is 180°.
Exterior Angle
P
1
2
Q
3
4
R
P
1
2
Q
3
4
R
m1  m2  m3 180
m3  m4 180
m1 m2  m3  m3  m4
m1  m2  m4
The measure of the exterior angle of a
triangle is equal to the sum of the measure
of the two opposite interior angles.
Note: Homework Page 672 #37
1  60'
1'  60"
2813'46"  _______ 
38.24  ____  ____' ____"
DAY 2
Homework Questions
Page 667
#16
130
140
#15
2 3
6 7
60 1
4 5
8
11
9 10
12
70
#13
11
12
1
2
10
3
9
4
8
7
6
5
• Curve
• Curve
• Simple Curve
• Curve
• Closed Curve
•
•
•
•
Curve
Simple Curve
Closed Curve
Simple Closed Curve
• A simple closed curved divides the plane into 3
disjoint sets: The curve, the interior, and the
exterior.
Exterior
Interior
Jordan’s Curve Theorem
Jordan’s Curve Theorem
Jordan’s Curve Theorem
• Concave
• Convex
• Polygonal Curve
• Polygon – Simple, closed curve made up
of line segments. (A simple closed
polygonal curve.)
Classifying Polygons
• Polygons are classified according to the
number of sides.
Classifying Polygons
•
•
•
•
•
•
•
•
TRIANGLE – 3 sides
QUADRILATERAL – 4 sides
PENTAGON – 5 sides
HEXAGON – 6 sides
HEPTAGON – 7 sides
OCTAGON – 8 sides
NONAGON – 9 sides
DECAGON – 10 sides
Classifying Polygons
• A polygon with n sides is
called an “n-gon”
• So a polygon with 20
sides is called a “20-gon”
Classifying Triangles
• According to the measure of the angles.
• According to the length of the sides.
Classifying Triangles
According to the measure of the angles.
• Acute Triangle: A triangle with 3 acute
angles.
• Right Triangle: A triangle with 1 right angle
and 2 acute angles.
• Obtuse Triangle: A triangle with 1 obtuse
angle and 2 acute angles.
Classifying Triangles
According to the length of the sides.
• Equilateral: All sides are congruent.
• Isosceles: At least 2 sides are congruent.
• Scalene: None of the sides are congruent.
ABC
ACD
ACE
ADE
AEF
CDE
B
C
D
A
F
E
B
ABC
ACD
ACE
ADE
AEF
CDE
C
D
A
Obtuse, Isosceles
Right , Scalene
Acute, Equilateral
Right , Scalene
Obtuse, Isosceles
Obtuse, Isosceles
F
E
Classifying Quadrilaterals
• Trapezoid – Quadrilateral with at least one pair
of parallel sides.
Quadrilaterals
Trapezoids
Classifying Quadrilaterals
• Trapezoid – Quadrilateral with at least one pair
of parallel sides.
• Parallelogram – A Quadrilateral with 2 pairs of
parallel sides.
Quadrilaterals
Trapezoids
Parallelograms
Classifying Quadrilaterals
• Trapezoid – Quadrilateral with at least one pair
of parallel sides.
• Parallelogram – A Quadrilateral with 2 pairs of
parallel sides.
• Rectangle – A Quadrilateral with 2 pairs of
parallel sides and 4 right angles.
Quadrilaterals
Trapezoids
Parallelograms
Rectangles
Classifying Quadrilaterals
• Trapezoid – Quadrilateral with at least one pair
of parallel sides.
• Parallelogram – Quadrilateral with 2 pairs of
parallel sides.
• Rectangle – Quadrilateral with 2 pairs of
parallel sides and 4 right angles.
• Rhombus – Quadrilateral with 2 pairs of
parallel sides and 4 congruent sides.
Quadrilaterals
Trapezoids
Parallelograms
Rectangles
Rhombus
Classifying Quadrilaterals
• Trapezoid – Quadrilateral with at least one pair
of parallel sides.
• Parallelogram – Quadrilateral with 2 pairs of
parallel sides.
• Rectangle – Quadrilateral with 2 pairs of
parallel sides and 4 right angles.
• Rhombus – Quadrilateral with 2 pairs of parallel
sides and 4 congruent sides.
• Square – Quadrilateral with 2 pairs of parallel
sides, 4 right angles, and 4 congruent sides.
Quadrilaterals
Trapezoids
Parallelograms
Rectangles
Square
Rhombus
• Equilateral
– All sides are congruent
• Equiangular
– Interior angles are congruent
Figure 11.20, Page 689
• Regular Polygons are equilateral and
equiangular.
• Interior Angles
• Interior Angles
• Exterior Angles – The sum of the measures of
the exterior angles of a polygon is 360°.
• Interior Angles
• Exterior Angles
• Central Angles – The sum of the measure of the
central angles in a regular polygon is 360°.
• Interior Angles
• Exterior Angles
• Central Angles
Classifying Angles Lab
Day 3
• Circle
• Compass
• Center
center
• Radius
radius
• Chord
radius
chord
• Diameter
diameter
radius
chord
• Circumference
circumference
diameter
radius
chord
• Tangent
circumference
diameter
radius
chord
tangent
•
•
•
•
•
•
•
•
Circle
Compass
Center
Radius
Chord
Diameter
Circumference
Tangent
circumference
diameter
radius
chord
tangent
Find and Identify
1.
3.
5.
7.
9.
11.
13.
E
I
C
B
D
G
L
2.
4.
6.
8.
10.
12.
K
A
M
J
F
H
C
E
H
D
F
I
A
B
DB  BG
CDE  EDB
G
Classifying Angles Lab
t
l
m n
ln
m
13 12
14
9 10 11
n
8 7
1 2
6
5
3
4
What’s Inside?
How do you find the sum of the
measure of the interior angles
of a polygon?
Example 11.8
Page 679
N
z
8x
E
y 3x
P
8x
8x
T
3x
A
Example 11.9
Page 680
x
3x
3x
x
x
3x
3x
3x
x
x
Classifying Quadrilaterals
and
Geo-Lingo Lab
Day 4
Make a Square!
Tangrams – Ancient Chinese Puzzle
Tangrams, 330 Puzzles, by Ronald C. Read
Sir Cumference Books
• Sir Cumference and the First Round Table
by Cindy Neuschwander
Also:
• Sir Cumference and the Great Knight of
Angleland
• Sir Cumference and the Dragon of Pi
• Sir Cumference and the Sword Cone
Angle Practice
me  66
mf  114
mg  75
mh  105
mi  105
mj  10
mk  85
ml  56
mm  37
38
j
95
g h
k
i
75
l
f e
m
39
mh  95
mj  53
mi  95
mk  53
ml  33
mm  85
mn  64
mp  65
mo  84
mq  65
mr  85
mt  85
ms  33
mu  62
mv  62
mw  86
mx  32
l
30
94
r
w
62 s
t
h i
m
23
31
q k
u
v
j p
32
n o
x
31
Must – Can’t – May Answers
Homework Questions
Page 688
#22
• Space
• Half Space
• A plane separates space into 3 disjoint
sets, the plane and 2 half spaces.
• Parallel Planes
• Dihedral Angle
• Points of Intersection
• If two planes intersect, their intersection is
a line.
• Simple Closed Surface
Figure 11.26, Page 698
• Solid
• Sphere
• Convex/Concave
Polyhedron
• A POLYHEDRON (plural - polyhedra) is a
simple closed surface formed from planar
polygonal regions.
• Edges
• Vertices
• Faces
• Lateral Faces – Page 699
• Prism
• Pyramid
• Apex
• Cylinder
• Cone
• Apex
• Right Prisms, Pyramids, Cylinders and
Cones
• Oblique Prisms, Pyramids, Cylinders and
Cones
Regular Polyhedron
• A three-dimensional figure whose faces
are polygonal regions is called a
POLYHEDRON (plural - polyhedra).
• A REGULAR POLYHEDRON is one in
which the faces are congruent regular
polygonal regions, and the same number
of edges meet at each vertex.
Regular Polyhedron
• Polyhedron made up of congruent regular
polygonal regions.
• There are only 5 possible regular
polyhedra.
Make Mine Platonic
Regular
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
n-gon
Number of Sides
Sum of Interior
Angles
Measure of each
interior angle
Make Mine Platonic
Regular
Polygon
Number of Sides
Sum of Interior
Angles
Measure of each
interior angle
Triangle
3
4
5
6
7
8
180°
360°
540°
720°
900°
1080°
60°
90°
108°
120°
128 4/7°
135°
n
(n - 2)180
(n-2)180/n
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
n-gon
• As the number of sides of a regular
polygon increases, what happens to the
measure of each interior angle? __
• Because they are formed from regular
polygons, our search for regular polyhedra
will begin with the simplest regular
polygon, the equilateral triangle.
• Each angle in the equilateral triangle
measures _____.
• Use the net with 4 equilateral triangles to
make a polyhedron.
• To make a three-dimensional object, we
need to engage 3 planes. Therefore, we
begin with three triangles at each vertex.
• What is the sum of the measures of the
angles at any given vertex? __
• This regular polyhedron is called a
TETRAHEDRON. A tetrahedron has __
faces. Each face is an __ __. We made
this by joining __ __ at each vertex.
• Form a polyhedron with the net that has 8
equilateral triangles. You will join 4
triangles at each vertex.
• What is the sum of the measure of the
angles at any given vertex? __
• This regular polyhedron is called an
OCTAHEDRON. An octahedron has __
faces. Each face is an __ __. At each
vertex, there are __ __.
• Use the net with 20 equilateral triangles to
form a polyhedron. You will join 5 triangles
at each vertex.
• What is the sum of the measure of the
angles at any given vertex? __
• This regular polyhedron is called an
ICOSAHEDRON. An icosahedron has
__ faces. Each face is an __ __. At each
vertex, there are __ __.
• When we join 6 equilateral triangles at a vertex,
what happens? Can you make a polyhedron
with 6 equilateral triangles at a vertex? __
• Is it possible to put more than 6 equilateral
triangles at a vertex to form a polyhedron? __
• Name the only three regular polyhedra that can
be made using congruent equilateral triangles:
__ __ __
• A regular quadrilateral is most commonly
known as a __.
• Each angle in the square measures __.
• Use the net with squares to make a
polyhedron.
• To make a three-dimensional object, we
need to engage 3 planes. Therefore, we
begin with three squares at each vertex.
• What is the sum of the measures of the
angles at any given vertex? __
• This regular polyhedron is called a
HEXAHEDRON. A hexahedron has __
faces. Each face is a __. At each vertex,
there are __ __.
• When we join 4 squares at a vertex, what
happens? Can you make a polyhedron
with 4 squares at a vertex? __
• Is it possible to put more than 4 squares at
a vertex to form a polyhedron? __
• Name the only regular polyhedron that can
be made using congruent squares. __
• A five-sided regular polygon is called a __.
• Each interior angle measures __.
• Use net with regular pentagons to make a
polyhedron. To make a three-dimensional
object, we need to engage 3 planes.
Therefore, we begin with three pentagons
at each vertex.
• What is the sum of the measures of the
angles at any given vertex? __
• This regular polyhedron is called a
DODECAHEDRON. A dodecahedron has
__ faces. Each face is a __. At each
vertex, there are __ __.
• Is it possible to put 4 or more pentagons at
a vertex and still have a three-dimensional
object? __
• Name the only regular polyhedron that can
be made using congruent pentagons. __
• A six-sided regular polygon is called a __.
• Each interior angle measures __.
• Is it possible to put 3 or more hexagons at
a vertex and still have a three-dimensional
object? __
• Is it possible to use any regular polygons
with more than six sides together to form a
regular polyhedron? __
(Refer to the table on page one for
numbers to verify)
• Only five possible regular polyhedra exist.
The union of a polyhedron and its interior
is called a “solid.” These five solids are
called PLATONIC SOLIDS.
Regular
Polyhedron
Number
Each
Number of
of Faces Face is a Polygons at
a vertex
Regular
Polyhedron
Number
Each
Number of
of Faces Face is a Polygons at
a vertex
Tetrahedron
4
Triangle
3
Octahedron
8
Triangle
4
Icosahedron
20
Triangle
5
Hexahedron
6
Square
3
Dodecahedron
12
Pentagon
3
Day 5
Homework Questions
Page 709
#29
Konigsberg Bridge Problem
C
A
B
D
C
A
B
D
Networks
A network consists of vertices – points in a
plane, and edges – curves that join some
of the pairs of vertices.
Traversable
A network is traversable if you can trace
over all the edges without lifting your
pencil.
A
B
A
C
B
D
C
B
C
A
B
A
D
B
A
B
C
D
D
A
A
A
C
C
D
B
C
B
C
E
A
A
B
B
F
F
D
C
D
E
B
A
D
A
E
F
C
B
C
D
C
Konigsberg Bridge Problem
C
A
B
D
A
O
B
D
E
F
C
G
B
A
O
D
E
F
G
C
B
A
O
D
E
F
G
C
The network is traversable.
Skit-So Phrenia!
Seeing the Third Dimension
2
1
1
3
1 2
Day 6
Homework Questions
Page 722
#27
#28
11x
x
11x
13x
13x
11x
13x
7x
4x
#29
y
x
75
150
z
Topology
• Topology is a study which concerns itself
with discovering and analyzing similarities
and differences between sets and figures.
• Topology has been referred to as “rubber
sheet geometry”, or “the mathematics of
distortion.”
Euclidean Geometry
• In Euclidean Geometry we say that two
figures are congruent if they are the exact
same size and shape.
• Two figures are said to be similar if they
are the same shape but not necessarily
the same size.
Topologically Equivalent
Two figures are said to be topologically
equivalent if one can be bended,
stretched, shrunk, or distorted in such a
way to obtain the other.
Topologically Equivalent
A doughnut and a
coffee cup are
topologically
equivalent.
According to Swiss psychologist Jean
Piaget, children first equate geometric
objects topologically.
Mobius Strip
We will consider 3 attributes that any two
topologically equivalent objects will share:
• Number of sides
• Number of edges
• Number of punctures or holes
Consider one strip of paper
• How many sides does it have?
• How many edges does it have?
Consider one strip of paper
• How many sides does it have? 2
• How many edges does it have? 1
Now make a loop with the strip of paper and
tape the ends together.
• How many sides does it have?
• How many edges does it have?
Now make a loop with the strip of paper and
tape the ends together.
• How many sides does it have? 2
• How many edges does it have? 2
Now cut the loop in half down the center of
the strip. Describe the result.
Mobius Strip
This time make a loop but before taping the
ends together, make a half twist. This is
called a Mobius Strip.
• How many sides does it have?
• How many edges does it have?
Mobius Strip
This time make a loop but before taping the
ends together, make a half twist. This is
called a Mobius Strip.
• How many sides does it have? 1
• How many edges does it have? 1
Now cut the Mobius strip in half down the
center of the strip. Describe the result.
• How many sides does your result have?
• How many edges?
• How many sides does your result have? 2
• How many edges? 2
• What do you think will happen if we cut the
resulting strip in half down the center?
• Try it! What happened?
• Make another Mobius strip
• Draw a line about 1/3 of the distance from
the edge through the whole strip.
• What do you think will happen if we cut on
this line?
• Try it! What happened?
• Use your last two strips to make two
untwisted loops, interlocking.
• Make sure they are taped completely
• Tape them together at a right angle. (They
will look kind of like a 3 dimensional 8.)
• Cut both strips in half lengthwise.
Did you know that 2 circles make a square?
• Compare the number of sides and edges
of the strip of paper, the loop, and the
Mobius strip.
• Are any of those topologically equivalent?