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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
• Line
• Collinear
• Plane
A
CB< > BC
D CB< >
G
F
E
• 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.
• Line segment
• Endpoint
• Length
HIIH
• Congruent
JK LM
m(JK) = m(LM)
JK LM
M
L
K
J
• Midpoint
NO OP
PON
• Half Line
• A point separates a line into 3 disjoint sets:
The point, and 2 half lines.
BXA< >
• Ray - the union of a half line and the point.
STTS>
• Angle – the union of two rays with a common endpoint.
UWVVWUWW
VU
• Vertex: W Common endpoint of the two rays.
• Sides: WU and WV
Exterior
Interior
• The angle separates the plane into 3 disjoint sets: The angle, the interior of the angle, and the exterior of the angle.
Exterior
Interior
• 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
1
23
45
6
Adjacent Angles
12
3
456
Vertical Angles
1
23
456
Vertical Angles
12 3
456
1
2
The sum of the measures of Complementary The sum of the measures of Complementary Angles is 90°.Angles is 90°.
• Complementary angles
• Adjacent complementary angles
3060
1 2
The sum of the measures of The sum of the measures of Supplementary Angles is 180°.Supplementary Angles is 180°.
• Supplementary Angles
• Adjacent Supplementary Angles
30
150
• Lines cut by a Transversal – these lines are not concurrent.
• Transversal• Corresponding Angles
1 2
34
5 6
78
• Transversal• Corresponding Angles
12
34
5 6
78
• Parallel lines Cut by a Transversal
• Parallel lines Cut by a Transversal• Corresponding Angles
1 2
34
5 6
78
• Parallel lines Cut by a Transversal• Corresponding Angles
1
2
34
5 6
78
• Describe the relative position of angles 3 and 5.• What appears to be true about their measures?
1 2
34
5 6
78
• Alternate Interior Angles
1 2
34
5 6
78
• Describe the relative positions of angles 1 and 7.• What appears to be true about their measures?
1 2
34
5 6
78
• Alternate Exterior Angles
1 2
34
5 6
78
Triangle
The sum of the measure of the interior angles of any triangle is 180°.
Exterior Angle
432
1
RQ
P
432
1
RQ
P
18043
180321
mm
mmm
43321 mmmmm
421 mmm
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
____"____'____24.38
_______"46'1328
"60'1'601
DAY 2
Homework QuestionsPage 667
#16
140
130
#15
701211
1098
765432160
#13 11
10
8
7 5
4
2
1
9
6
3
12
• 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.
F E
D
CB
A
CDE
AEF
ADE
ACE
ACD
ABC
F E
D
CB
A
IsoscelesObtuseCDE
IsoscelesObtuseAEF
ScaleneRightADE
lEquilateraAcuteACE
ScaleneRightACD
IsoscelesObtuseABC
,
,
,
,
,
,
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.
Square
RhombusRectangles
Parallelograms
Trapezoids
Quadrilaterals
• 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
chord
radius
• Diameter
diameter
chord
radius
• Circumference
circumference
diameter
chord
radius
• Tangent
tangent
radius
chord
diameter
circumference
• Circle• Compass• Center• Radius• Chord• Diameter• Circumference• Tangent
tangent
radius
chord
diameter
circumference
Find and Identify
1. E 2. K
3. I 4. A
5. C 6. M
7. B 8. J
9. D 10. F
11. G 12. H
13. L
DB BGCDE EDBIH
GF
E
D
C
B
A
Classifying Angles Lab
1413 12
11109
8 7 6 54321
m nl n
tl
n
m
What’s Inside?
How do you find the sum of the measure of the interior angles
of a polygon?
Example 11.8Page 679
z8x
8x8x
3x 3xy
A
T
N
E
P
Example 11.9Page 680
3x
3x3x
3x
3x
x
x x
x
x
Classifying Quadrilateralsand
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
37
56
85
10
105
105
75
114
66
mm
lm
km
jm
im
hm
gm
fm
em
ml
k
j
ihg
f e 39
38
75
95
32
8662
6285
3385
6565
8464
8533
5353
9595
xm
wmvm
umtm
smrm
qmpm
omnm
mmlm
kmjm
imhm
62
30
94
31
32
3123
x
w
vut
s
r
q
p
on
m
l
k
jih
Must – Can’t – May Answers
Homework QuestionsPage 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
• 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.
Regular Polyhedron
Make Mine Platonic
Regular
Polygon
Number of Sides Sum of Interior Angles
Measure of each interior angle
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
n-gon
Make Mine Platonic
Regular
Polygon
Number of Sides Sum of Interior Angles
Measure of each interior angle
Triangle 3 180° 60°Quadrilateral 4 360° 90°
Pentagon 5 540° 108°Hexagon 6 720° 120°Heptagon 7 900° 128 4/7°Octagon 8 1080° 135°
n-gon n (n - 2)180 (n-2)180/n
• 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
of Faces
Each
Face is a
Number of Polygons at
a vertex
Regular
Polyhedron
Number
of Faces
Each
Face is a
Number of 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 QuestionsPage 709
#29
Konigsberg Bridge Problem
D
C
B
A
D
C
B
A
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.
CB
A A
B C CB
A BA
D C
CD
A BBA
D C
CD
A B
CD
A B
F
E
CD
A B F
E
D C
B
A
FE
D
C
BA
CD
A B
Konigsberg Bridge Problem
D
C
B
A
O
GFE
D
C
BA
O
GFE
D
C
BA
O
GFE
D
C
BA
The network is traversable.
Skit-So Phrenia!
Seeing the Third Dimension
2
3
11
1
2
Day 6
Homework QuestionsPage 722
#27
#28
11x11x
13x 13x 13x11x 7x
4xx
#29
z
y x
150
75
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?