geometry chapter 11. informal study of shape until about 600 b.c. geometry was pursued in response...

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

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

• 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.


• TRIANGLE – 3 sides• QUADRILATERAL – 4 sides• PENTAGON – 5 sides• HEXAGON – 6 sides• HEPTAGON – 7 sides• OCTAGON – 8 sides• NONAGON – 9 sides• DECAGON – 10 sides


• 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.


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.


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



• Parallelogram – A Quadrilateral with 2 pairs of parallel sides.

Quadrilaterals

Trapezoids

Parallelograms



• 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

• 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

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

• 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.


• 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.


• 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

Konigsberg Bridge Problem

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

#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.



Now make a loop with the strip of paper and tape the ends together.



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.



Mobius Strip

This time make a loop but before taping the ends together, make a half twist. This is called a Mobius Strip.



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?

geometry chapter 11. informal study of shape until about 600 b.c. geometry was pursued in response...

Documents

transversal slide

concurrent slide

congruent slide

midpoint slide

reflex angle slide

degrees protractor slide

adjacent angles

parallel lines