Chapter 9 Geometry

1

• Chapter 9 Geometry
• 9.1 Points, Lines, Planes, and Angles
• 9.2 Polygons
• 9.3 Perimeter and Area
• 9.4 Volume
• 9.5 The Möbius Strip, Klein Bottle, and Maps
• 9.6 Non-Euclidean Geometry

2

• 9.1 Points, Lines, Planes, and Angles
• One of the first steps in learning geometry is
  learning the terms and definitions of geometry.
  This section covers these terms and definitions.

3

• Euclidean Geometry - History
• In approx 300 B.C., Euclid summarized much of the
  Greek mathematics of his time. He summarized it
  in his greatest work, The Elements. The Elements
  was a 13 book set which laid the foundation for
  plane geometry, called Euclidean geometry.
• Interestingly, these books were studied by
  Abraham Lincoln when he was in law school.
•  

4

• Euclidean Geometry - History
• Euclid was the first mathematician of his time to
  use the axiomatic method.
• Euclids axiomatic system consisted of four
  parts undefined terms, which lead to
  definitions, which lead to postulates (also
  called axioms, which are accepted as true), which
  lead to theorems (which are proven by deductive
  reasoning).

5

• Euclidean Geometry History
• The undefined terms on which Euclid based his
  system were point, line, and plane.
• Point a location in space
• Line a straight arrangement of points
• Plane a two dimensional surface that extends
  infinitely in both directions ( i.e., a table
  top )
• Euclid used the undefined terms to introduce
  certain definitions as they were needed in his
  axiomatic system.

6

• Euclidean Geometry Definitions
• half line a piece of a line containing all the
  points lying to one side of a given endpoint.
• ray a piece of a line which contains an
  endpoint and all the points lying to one side of
  the endpoint.
• line segment a piece of a line with two
  endpoints
• Notice how in the definition of half line, an
  endpoint is not included. These, as well as open
  line segment and half open line segment, are
  illustrated in Figure 9.2 on page 432 in the
  text. Note the diagram and the symbols for the
  different terms.

7

• Euclidean Geometry Definitions
• Remember intersection and union? These are used
  again in geometry. Look at Example 1 on page
  433-434 in the text. This example illustrates
  how to determine the solution to different
  intersections and unions of lines, rays, half
  lines, etc. Pay special attention to the
  symbols, especially the half line.

8

• Euclidean Geometry Definitions
• Remember that a plane is a two dimensional
  surface that extends infinitely in both
  directions. Some properties of planes can be
  found in Figures 9.4 and 9.5 on page 434 in the
  text. Planes are very important to some of
  Euclids definitions.
• parallel lines are two lines, lying in a plane
  that do not intersect.
• parallel planes are two planes that remain the
  same distance apart.
• skew lines are two lines that do not intersect
  and do not lie on the same plane.
•  

9

• Euclidean Geometry Definitions
• angle the union of two rays with a common
  endpoint (? ).
• vertex the common endpoint of the two rays.
• sides of an angle the rays that make up an
  angle.
• More about angles can be found in Figure 9.6 on
  page 434 in the text. Example 2 on page 435 of
  the text illustrates how to find unions and
  intersections that involve angles and rays.

10

• Euclidean Geometry Definitions
• measure of an angle the size of an angle
  measured by the amount of rotation needed to
  turn one side of the angle to the other by
  pivoting about the vertex.
• Angles are measured in degrees ( ? ).
• protractor a device used to measure angles.

11

• Euclidean Geometry Definitions
• right angle an angle that measures 90?.
• acute angle an angle that measures less than
  90?.
• obtuse angle an angle that measures more than
  90?, but less than 180?.
• straight angle an angle that measures 180?.

obtuse angle
right angle
acute angle
straight angle
12

• Euclidean Geometry Definitions
• adjacent angles two angles that have a common
  vertex and side, but no common interior points
  (i.e.,?1?2, ?2?3,? 3?4, ?4?1).
• vertical angles a pair of nonadjacent angles
  formed by two intersecting lines (i.e., ?1?3 ,
  ?2?4). Vertical angles are congruent in
  measure.
•  

13

• Euclidean Geometry Definitions
• complementary angles two angles whose sum
  measures 90?.
• supplementary angles two angles whose sum
  measures 180?.

1
3
4
2
complementary angles
supplementary angles
14

• Euclidean Geometry Definitions
• transversal a line that intersects two
  different lines at two different points.
• When two parallel lines are cut by a transversal,
  eight angles are formed. (lines m n, below, are
  parallel)

15

• Euclidean Geometry Definitions
• When two parallel lines are cut by a transversal,
  interior and exterior angles are formed.

given lines m and n are parallel.
16

• Euclidean Geometry Definitions
• When two parallel lines are cut by a transversal,
  interior and exterior angles are formed.
• interior angles angles that lie between the
  parallel lines (?3,?4,?5,?6).

given lines m and n are parallel.
17

• Euclidean Geometry Definitions
• When two parallel lines are cut by a transversal,
  interior and exterior angles are formed.
• interior angles angles that lie between the
  parallel lines (?3,?4,?5,?6).
• exterior angles angles that lie outside the
  parallel lines
  (?1,? 2,?7, ?8).

given lines m and n are parallel.
18

• Euclidean Geometry Definitions
• alternate interior angles (AIA) interior angles
  on opposite sides of the transversal.
• alternate exterior angles (AEA) exterior
  angles on opposite sides of the transversal.
• corresponding angles (CA) - one interior and one
  exterior angle on the same side of a
  transversal. They are located in the same
  place in each group of four angles.
• When two parallel lines are cut by a transversal,
  AIA, AEA, and CA are congruent in measure.  

19

• Euclidean Geometry Definitions

alternate interior angles (AIA) ?3?6,
?5?4. alternate exterior angles (AEA) ?2?7,
?1?8. corresponding angles (CA) ?1?5, ?2?6,
?3?7, ?4?8.
given lines m n are parallel
20

• Euclidean Geometry Definitions
• Alternate interior angles, alternate exterior
  angles, and corresponding angles are also
  summarized at the bottom of page 437 in the text.
• An important thing to remember is that alternate
  interior angles, alternate exterior angles, and
  corresponding angles are congruent (the same
  measure) only when the two lines are parallel.
  If you are not told that the lines are parallel,
  then the measures of the angles may not be
  congruent.

21

• Angle Diagram 1

Determine the measures of angles 1 8 in the
following diagram, given that angle 6 measures
70?. Assume lines m n are parallel.
22

• Angle Diagram 1

Replace angle 6 with 70? .
23

• Angle Diagram 1

There are many different ways to solve this
problem. Lets find the measure of angle 7 next.
24

• Angle Diagram 1

The measure of angle 7 is 70?, since vertical
angles are congruent.
25

• Angle Diagram 1

The measure of angle 8 is 110? since 180 70
110. (supplementary angles)
26

• Angle Diagram 1

The measure of angle 5 is 110? since vertical
angles are congruent.
27

• Angle Diagram 1

The measure of angle 1 is 110? since
corresponding angles are congruent.
110?
28

• Angle Diagram 1

The measure of angle 2 is 70? since 180 110
70. (supplementary angles)
110?
29

• Angle Diagram 1

The measure of angle 3 is 70? by vertical angles.
110?
30

• Angle Diagram 1

The measure of angle 4 is 110? because 180 70
110. (supplementary angles)
110?
31

• Angle Diagram 1

The diagram is complete.
110?
32

• Angle Diagram 2

The following angles are complementary angles.
Find the measures of ? 1 ? 2.
33

• Angle Diagram 2

Since the angles are complementary, ? 1 ? 2
90?.
34

• Angle Diagram 2

Since the angles are complementary, ? 1 ? 2
90?. x 4x-10 90?
35

• Angle Diagram 2

• Since the angles are complementary, ? 1 ? 2
  90?.
• x 4x-10 90?
• 5x 10 90?
• 5x 100?
• x 20?

36

• Angle Diagram 2

• Since the angles are complementary, ? 1 ? 2
  90?.
• x 4x-10 90?
• 5x 10 90?
• 5x 100?
• x 20?
• 1 20?
• 2 4(20) 10
• 80 10 70?

37

• Modeling Angles 3
• The difference between the measures of two
  complementary angles is 24?. Determine the
  measures of the two angles.

38

• Modeling Angles 3
• The difference between the measures of two
  complementary angles is 24?. Determine the
  measures of the two angles.
• Define the variables
• let x the larger angle
• let y the smaller angle

39

• Modeling Angles 3
• The difference between the measures of two
  complementary angles is 24?. Determine the
  measures of the two angles.
• Define the variables
• let x the larger angle
• let y the smaller angle
• Write the 1st equation
• x y 90
• Sums to 90 since the two angles are
  complementary.

40

• Modeling Angles 3
• Write the 2nd equation
• x y 24
• The difference of the two angles is 24?.

41

• Modeling Angles 3
• Write the 2nd equation
• x y 24
• The difference of the two angles is 24?.
• The system of equations
• x y 90
• x y 24

Use the addition method. Add down.
42

• Modeling Angles 3
• Write the 2nd equation
• x y 24
• Since the difference of the two angles is 24?.
• The system of equations
• x y 90
• x y 24
• 2x 114

Use the addition method. Add down.
43

• Modeling Angles 3
• Write the 2nd equation
• x y 24
• Since the difference of the two angles is 24?.
• The system of equations
• x y 90
• x y 24
• 2x 114
• x 57

Use the addition method. Add down.
44

• Modeling Angles 3
• Write the 2nd equation
• x y 24
• Since the difference of the two angles is 24?.
• The system of equations
• x y 90
• x y 24
• 2x 114
• x 57

Now substitute 57 into the first equation.

45

• Modeling Angles 3
• Write the 2nd equation
• x y 24
• Since the difference of the two angles is 24?.
• The system of equations
• x y 90
• x y 24
• 2x 114
• x 57

Now substitute 57 into the first equation.

57 y 90, so y 33.
46

• Modeling Angles 3
• Write the 2nd equation
• x y 24
• Since the difference of the two angles is 24?.
• The system of equations
• x y 90
• x y 24
• 2x 114
• x 57

Now substitute 57 into the first equation.

57 y 90, so y 33. The solution is x 57?,
y 33?
47

• Modeling Angles 4
• If ?1 ?2 are supplementary angles and ?1 is
  eight times as large as ?2,find the measures of
  ?1 ?2.

48

• Modeling Angles 4
• If ?1 ?2 are supplementary angles and ?1 is
  eight times as large as ?2,find the measures of
  ?1 ?2.
• Define the variables
• let x measure ?2
• let 8x measure ?1

49

• Modeling Angles 4
• If ?1 ?2 are supplementary angles and ?1 is
  eight times as large as ?2,find the measures of
  ?1 ?2.
• Define the variables
• let x measure ?2
• let 8x measure ?1
• Write the equation
• 8x x 180

50

• Modeling Angles 4
• If ?1 ?2 are supplementary angles and ?1 is
  eight times as large as ?2,find the measures of
  ?1 ?2.
• Define the variables
• let x measure ?2
• let 8x measure ?1
• Write the equation
• 8x x 180
• Solve the equation
• 9x 180
• x 20

51

• Modeling Angles 4
• If ?1 ?2 are supplementary angles and ?1 is
  eight times as large as ?2,find the measures of
  ?1 ?2.
• Define the variables
• let x measure ?2
• let 8x measure ?1
• Write the equation
• 8x x 180
• Solve the equation
• 9x 180
• x 20

Thus, ?2 20?, and ?1 8(20) 160?.
52

• 9.2 - Polygons
• This section covers polygons, which are figures
  that are made up of line segments. Many of the
  theorems in geometry have to do with polygons.

53

• Polygon Definitions
• polygon a closed planar figure created by
  three or more straight segments.
• sides of a polygon the straight segments that
  form a polygon.
• vertices of a polygon (singular, vertex) the
  points where two sides meet in a polygon.
• polygonal region the union of the sides of a
  polygon and its interior.

a polygon
54

• Polygon Definitions
• equilateral polygon a polygon whose sides are
  the same length.
• equiangular polygon a polygon whose angles are
  equal in measure.
• regular polygon a polygon whose sides are the
  same length and whose angles are equal in
  measure.
•  

55

• Polygon Definitions
• Polygons are named according to their number of
  sides. Table 9.1 on page 441 in the text lists
  the names of different polygons, and their
  corresponding number of sides.

56

• Triangles
• One of the most important polygons is the
  triangle. An important thing to know about
  triangles is that the sum of the measures of the
  interior angles of a triangle is 180 degrees.

Thus, the missing angle in this triangle would be
105?, since 30 45 105 180.
45
30
57

• Angle Diagram - Triangles

1. Find the measure of ?x.
58

• Angle Diagram - Triangles

1. Find the measure of ?x.
This diagram is basically a triangle. If we can
find the other two angles of the triangle, then
we can subtract them from 180 to find ?x.
59

• Angle Diagram - Triangles

1. Find the measure of ?x.
This angle measures 25? because vertical angles
are congruent.
25?
60

• Angle Diagram - Triangles

1. Find the measure of ?x.
This angle measures 138? because vertical angles
are congruent.
61

• Angle Diagram - Triangles

1. Find the measure of ?x.
Now we know the measure of two angles of a
triangle. We can calculate the 3rd angle.
62

• Angle Diagram - Triangles

1. Find the measure of ?x.
25 138 x 180 163 x 180 x
17?
63

• Angle Diagram - Triangles

1. Find the measure of ?x.
25 138 x 180 163 x 180 x
17?
64

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
2
1
3
r
4
5
6
67?
7
8
s
9
11
12
10
65

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?9 67?, since vertical angles are congruent.
2
1
3
r
4
5
6
67?
7
8
s
67?
11
12
10
66

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?6 180 67 113?.
r
s
67

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?10 113?, by vertical angles.
r
s
68

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?4 67?, since alternate interior angles are
congruent.
r
s
69

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?3 67?, by vertical angles.
r
s
70

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?2 63?, by vertical angles.
r
s
71

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?1 180 63 67 ?1 50?.
r
s
72

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?5 50?, by vertical angles.
r
s
73

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?7 50?, by alternate interior angles.
r
s
74

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?12 50?, by vertical angles.
r
s
75

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?8 180 50 130?.
r
s
76

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
?11 130?, by vertical angles.
r
s
77

• Angle Diagram - Triangles

2. Find the measures of ?1 - ?12. Lines r and s
are parallel.
The diagram is complete!
r
s
78

• Interior angles of a polygon
• The sum of the measures of the interior angles of
  an n-sided polygon is (n-2)180.
• This formula is based on the idea that the
  interior angles of a triangle sum to 180?.

79

• Interior angles of a polygon
• The sum of the measures of the interior angles of
  an n-sided polygon is (n-2)180.
• This formula is based on the idea that the
  interior angles of a triangle sum to 180?.
• Given a pentagon

80

• Interior angles of a polygon
• The sum of the measures of the interior angles of
  an n-sided polygon is (n-2)180.
• This formula is based on the idea that the
  interior angles of a triangle sum to 180?.
• Given a pentagon
• Draw in the diagonals from one vertex.

81

• Interior angles of a polygon
• The sum of the measures of the interior angles of
  an n-sided polygon is (n-2)180.
• This formula is based on the idea that the
  interior angles of a triangle sum to 180?.
• Given a pentagon
• Draw in the diagonals from one vertex.
• 3 triangles are formed.
• 3 x 180 540

82

• Interior angles of a polygon
• The sum of the measures of the interior angles of
  an n-sided polygon is (n-2)180.
• This formula is based on the idea that the
  interior angles of a triangle sum to 180?.
• Given a pentagon
• Draw in the diagonals from one vertex.
• 3 triangles are formed.
• 3 x 180 540
• Using the formula, we find
• (5-2)180 3(180) 540

83

• Classification of Triangles
• Triangles are classified by angle measure and
  side length.
• Classification by angle
• Acute triangle a triangle with 3 acute angles.
• Obtuse triangle a triangle with 1 obtuse angle.
• Right triangle a triangle with 1 right angle.

84

• Classification of Triangles
• Classification by sides
• Isosceles triangle a triangle with 2 equal
  sides (and two equal angles).
• Equilateral triangle a triangle with 3 equal
  sides (and three equal angles).
• Scalene triangle a triangle where no two sides
  are equal in length (thus, no two angles are of
  equal length).

85

• Classification of Triangles
• At the top of page 443 in the text is a table
  illustrating triangles classified by their angles
  and by their sides. You will be asked to
  classify triangles in both ways on your homework
  (page 447 11-21).

86

• Similar Figures
• similar figures are figures that have the same
  shape, but may be of different sizes.
• Two polygons are similar if their corresponding
  angles have the same measure and their
  corresponding sides form a ratio.

87

• Similar Figures
• Similar figures can be any polygon, but most of
  the time, when we talk about similar figures, we
  are talking about similar triangles.
• Study Example 2 on page 443 in the text. It
  illustrates how to find the missing side of a
  triangle given a pair of similar triangles. Pay
  close attention to how the proportions are set up
  and solved.

88

• Example - Similar Figures

Find the length of side y, given that ? ABC and
? DEC are similar.
89

• Example - Similar Figures

First set up the proportion
90

• Example - Similar Figures

First set up the proportion
91

• Example - Similar Figures

First set up the proportion
We arrive at this proportion since 8 on the large
triangle corresponds with 6 on the small
triangle.
92

• Example - Similar Figures

First set up the proportion
y 3 on the large triangle corresponds with y on
the small triangle.
93

• Example - Similar Figures

Now solve the proportion
94

• Example - Similar Figures

Now solve the proportion
8y 6(y 3) 8y 6y 18 2y 18 y 9
95

• Example - Similar Figures

Now solve the proportion
8y 6(y 3) 8y 6y 18 2y 18 y 9
Thus, side y 9 inches.
96

• Congruent Figures
• congruent figures two similar figures whose
  corresponding sides are the same length.
• When figures are congruent, they are the exact
  same size, corresponding angles and corresponding
  sides are congruent.
• Read Example 4 on page 445 in the text. It is a
  example about sides and angles of congruent
  figures.

97

• Quadrilaterals
• quadrilaterals are four sided polygons.
• Since they have four sides, the sum of the
  interior angles of a quadrilateral is 360? (since
  (4-2)180 360).
• Read the chart in the middle of page 446 in the
  text. It summarizes quadrilaterals and
  classifies them according to their
  characteristics. Notice which definitions
  include parallel sides, congruent sides, and
  right angles.

98

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.

99

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 1. A trapezoid is a parallelogram.

100

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 1. A trapezoid is a parallelogram.
• False, a trapezoid only has one pair of opposite
  sides parallel, parallelograms have both pairs of
  opposite sides parallel.

101

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 2. A rhombus is a parallelogram.

102

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 2. A rhombus is a parallelogram.
• True, a rhombus has both pairs of opposite sides
  parallel.

103

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 3. A square is a rectangle.

104

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 3. A square is a rectangle.
• True, a rectangle is a parallelogram whose angles
  are right angles. A square is also a
  parallelogram whose angles are right angles.

105

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 4. A rectangle is a square.

106

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 4. A rectangle is a square.
• False, not only do squares have right angles, but
  the four sides of a square are equal in length.
  A rectangle does not necessarily have sides of
  equal length.

107

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 5. A rhombus is a square.
• False, not only do squares have four sides of
  equal length, but a square has four right angles.
  A rhombus does not necessarily have four right
  angles.

108

• Quadrilaterals
• After reviewing the chart on page 446 in the
  text, try to answer the following true/false
  statements.
• 6. A square is a rhombus.
• True, a rhombus is a parallelogram with four
  sides equal in length. A square is also a
  parallelogram with four sides equal in length.

109

• 9.3 - Perimeter and Area
• Geometric shapes exist in nature and in the world
  made by human beings. Being able to calculate
  the perimeter or area of a figure is useful when
  trying to build a fence, gutter a house, paint a
  house, even planting a garden. There is an
  unlimited number of situations where perimeter
  and area are useful.

110

• Perimeter
• The perimeter, P, of a two dimensional figure is
  the sum of the lengths of the sides of the
  figure.
• Perimeter is the distance around something, and
  since it is a distance, it is measured in feet,
  inches, meters, kilometers, etc.

111

• Perimeter
• The perimeter, P, of a two dimensional figure is
  the sum of the lengths of the sides of the
  figure.
• Perimeter is the distance around something, and
  since it is a distance, it is measured in feet,
  inches, meters, kilometers, etc.
• The perimeter of the trapezoid below would be the
  sum of the lengths of the 4 sides.

6 in
3 in
3 in
4 in
112

• Perimeter
• The perimeter, P, of a two dimensional figure is
  the sum of the lengths of the sides of the
  figure.
• Perimeter is the distance around something, and
  since it is a distance, it is measured in feet,
  inches, meters, kilometers, etc.
• The perimeter of the trapezoid below would be the
  sum of the lengths of the 4 sides.

6 in
Thus, P 6 3 4
3 16 in
3 in
3 in
4 in
113

• Area
• The area, A, is the region within the boundaries
  of the figure. Area is measured in square units,
  for example ft2, in2, km2.

114

• Area
• The area, A, is the region within the boundaries
  of the figure. Area is measured in square units,
  for example ft2, in2, km2.
• When calculating area, you want to figure out how
  many squared units it would take to tile the
  figure. The green color in the figure below
  represents what we are calculating when we find
  the area of the figure.

115

• Rectangle Formulas
• Given a rectangle
• Perimeter 2l 2w
• Area l x w

l
w
w
l
116

• Square Formula
• Given a square
• Area s2

s
s
s
s
117

• Parallelogram Formula
• Given a parallelogram
• Area b x h

Look at Figure 9.29 on page 451 in the text. See
how the parallelogram relates to a rectangle.
h
b
118

• Triangle Formula
• Given a triangle
• Area ½ bh

h
h
b
b
119

• Trapezoid Formula
• Given a trapezoid
• Area ½ h(b1 b2)

b1
h
b2
120

• Perimeter and Area Formulas
• Look at the perimeter and area summary chart at
  the bottom of page 452 in the text.
• It might be useful to write the formulas on a 3x5
  card. Use the card for a bookmark for this
  chapter, and then when you need the formulas, you
  can find them quickly. There will be more
  formulas in the next few sections of the text, so
  write small!
• Note When you take the test on this information,
  you can use your 3x5 card with the formulas
  written on it.

121

• Perimeter and Area
• 1. Find the area of the following figure.
  Dimensions are in meters.

10 m
11 m
9 m
9 m
6 m
122

• Perimeter and Area
• 1. Find the area of the following figure.
  Dimensions are in meters.

First write the formula Area ½
h(b1 b2)
10 m
11 m
9 m
9 m
6 m
123

• Perimeter and Area
• 1. Find the area of the following figure.
  Dimensions are in meters.

First write the formula Area ½
h(b1 b2) Then substitute in the lengths of the
sides and solve
10 m
11 m
9 m
9 m
6 m
124

• Perimeter and Area
• 1. Find the area of the following figure.
  Dimensions are in meters.

First write the formula Area ½
h(b1 b2) Then substitute in the lengths of the
sides and solve A
½(11)(10 6) A ½(11)(16) ½(176) A 88
m2
10 m
11 m
9 m
9 m
6 m
125

• The Pythagorean Theorem
• The Pythagorean theorem is an important tool when
  finding the areas and perimeters of triangles.
• The Pythagorean theorem states The sum of the
  squares of the lengths of the legs of a right
  triangle equals the square of the length of the
  hypotenuse.

a2 b2 c2
126

• The Pythagorean Theorem
• Study Example 2 on page 453 in the text. This
  problem illustrates how the Pythagorean theorem
  can be used to find the hypotenuse of a triangle.
• The example on the following slide shows how to
  use the Pythagorean theorem to find the length of
  one of the legs of a right triangle.

127

• The Pythagorean Theorem

Find the length of side x in the right triangle
below
128

• The Pythagorean Theorem

Find the length of side x in the right triangle
below
The Pythagorean theorem a2
b2 c2
129

• The Pythagorean Theorem

Find the length of side x in the right triangle
below
The Pythagorean theorem a2
b2 c2 x2 82 102 x2 64 100 x2 36 x
6 cm
130

• Circles
• A circle is a set of points that are equidistant
  from a fixed point called the center.
• A radius, r, of a circle is a line segment from
  the center of the circle to any point on the
  circle.
• A diameter, d, of a circle is a line segment
  through the center of a circle with both
  endpoints on the circle. The length of the
  diameter is twice the length of the radius.
• The circumference is the length of the simple
  closed curve that forms the circle.

131

• Circles

The ratio of the diameter of the circle to the
radius of the circle is approximately 3.14. We
call the ratio ?.
radius
diameter
132

• Circles

The ratio of the diameter of the circle to the
radius of the circle is approximately 3.14. We
call the ratio ?. In this course, use the ? key
on your calculator. If you do not have a ? key,
approximate ? 3.14.
radius
diameter
133

• Circles
• Other circle formulas
• The circumference of a circle, C 2?r or C ?d.

134

• Circles
• Other circle formulas
• The circumference of a circle, C 2?r or C ?d.
• The area of a circle, A ?r2

r
d
135

• Circles
• 2. Find the circumference and area of the circle
  below.

12 cm
136

• Circles
• 2. Find the circumference and area of the circle
  below.

C ?d ? (3.14)(12) ? 37.68 cm
137

• Circles
• 2. Find the circumference and area of the circle
  below.

C ?d ? (3.14)(12) ? 37.68 cm A ?r2
? (3.14)(6)2 ? (3.14)(36) ? 113.04 cm2
Note we used 6 since 6 is the radius of the
circle.
138

• Circles
• There is a nice illustration that summarizes all
  of the terms for circle in Figure 9.33 on page
  454 in the text.
• Study Examples 3 4 on pages 454 456 in the
  text. These examples illustrate the usefulness
  of the formulas in this section of the text.
• Study Example 5 on page 456 in the text. It is
  an important example because it illustrates how
  to convert areas to different units.

139

• Other Examples
• 3. Find the shaded area in the figure below.

18 cm
8 cm
6 cm
2 cm
140

• Other Examples
• 3. Find the shaded area in the figure below.

Many shading problems can be solved using
subtraction, since it is similar to taking a
large shape and cutting out a smaller shape.
18 cm
8 cm
6 cm
2 cm
141

• Other Examples
• 3. Find the shaded area in the figure below.

Many shading problems can be solved using
subtraction, since it is similar to taking a
large shape and cutting out a smaller shape.
In this example, find the area of the large
rectangle, and subtract the area of the small
rectangle.
18 cm
8 cm
6 cm
2 cm
142

• Other Examples
• 3. Find the shaded area in the figure below.

Area of large rectangle l x w 18 x 6 108
cm2 Area of small rectangle l x w 8 x 2 16
cm2
18 cm
8 cm
6 cm
2 cm
143

• Other Examples
• 3. Find the shaded area in the figure below.

Area of large rectangle l x w 18 x 6 108
cm2 Area of small rectangle l x w 8 x 2 16
cm2
108 cm2 - 16 cm2 92 cm2, thus the shaded area
equals 92 cm2.
18 cm
8 cm
6 cm
2 cm
144

• Other Examples
• 4. A picture frame 4 in. wide surrounds a
  portrait that is 11 in. wide by 17 in. high.
  Find the area of the picture frame.

145

• Other Examples
• 4. A picture frame 4 in. wide surrounds a
  portrait that is 11 in. wide by 17 in. high.
  Find the area of the picture frame.

Here is the picture.
11 in
17 in
146

• Other Examples
• 4. A picture frame 4 in. wide surrounds a
  portrait that is 11 in. wide by 17 in. high.
  Find the area of the picture frame.

Here is the frame.
11 in
17 in
147

• Other Examples
• 4. A picture frame 4 in. wide surrounds a
  portrait that is 11 in. wide by 17 in. high.
  Find the area of the picture frame.

The frame is 4 inches wide, but it actually adds
8 inches to the overall height and 8 inches to
the overall width.
11 in
17 in
148

• Other Examples
• 4. A picture frame 4 in. wide surrounds a
  portrait that is 11 in. wide by 17 in. high.
  Find the area of the picture frame.

19 in
Thus, the height of the picture frame is 25 in.
(17 8 25) and the width of the frame is 19 in
(11 8 19).
11 in
25 in
17 in
149

• Other Examples
• 4. A picture frame 4 in. wide surrounds a
  portrait that is 11 in. wide by 17 in. high.
  Find the area of the picture frame.

19 in
Work this as a subtraction problem.
11 in
25 in
17 in
150

• Other Examples
• 4. A picture frame 4 in. wide surrounds a
  portrait that is 11 in. wide by 17 in. high.
  Find the area of the picture frame.

19 in
25 x 19 475 in2 17 x 11 187 in2 475 187
288 in2
11 in
25 in
17 in
151

• Other Examples
• 4. A picture frame 4 in. wide surrounds a
  portrait that is 11 in. wide by 17 in. high.
  Find the area of the picture frame.

19 in
25 x 19 475 in2 17 x 11 187 in2 475 187
288 in2 The area of the picture frame is 288 in2.
11 in
25 in
17 in
152

• 9.4 Volume
• When working with a one-dimensional figure, we
  can find its length. When working with a
  two-dimensional figure, we can find its area.
  This section discusses three-dimensional figures.
  For a three-dimensional figure (i.e., a
  rectangular shoebox), we can find its volume.

153

• Volume
• Volume is the measure of capacity of a figure.
  Volumes are measured in cubic units such as cubic
  inches or cubic meters.
• Solid geometry is the study of three-dimensional
  solid figures. Volume and surface area can be
  calculated for these three-dimensional figures.
  This section covers volume.

154

• Volume Rectangular Solid
• Volume of a Rectangular solid

155

• Volume Cube
• A cube is a rectangular solid with the same
  length, width, and height.
• Volume of a cube

156

• Volume Cylinder
• Volume of a cylinder
• V?r2h

r
h
r
h
157

• Volume Cone
• Volume of a cone
• V1/3(?r2h)

158

• Volume Cone
• Volume of a cone
• V1/3(?r2h)

Look at Figure 9.36 on page 461 in the text.
Surprisingly enough, the cones volume is 1/3 the
volume of the cylinder that has the same base and
height.
159

• Volume Sphere
• Volume of a sphere
• V4/3(?r3)

160

• Volume Summary
• A summary chart of the preceding five volumes can
  be found at the bottom of page 461 in the text.
• Study Example 1 on page 462 in the text. It
  shows how to find the volume of a rectangular
  solid that relates to a real life situation.
  Notice how to determine how much the concrete
  would cost.
• Study Example 2 on page 462 in the text. It is
  an interesting problem having to do with
  cylinders.

161

• Polyhedrons
• A polyhedron is a closed surface formed by the
  union of polygonal regions.
• Each polygonal region is called a face of the
  polyhedron.
• The line segment formed by the intersection of
  two faces is called an edge.
• The point at which two or more edges intersect is
  called a vertex. (think corners)

edge
face
vertex
162

• Eulers Polyhedron Formula
• An interesting thing occurs when working with
  polyhedron.

163

• Eulers Polyhedron Formula
• An interesting thing occurs when working with
  polyhedron.

Think about what this means. Look at Figure 9.39
(c) on page 463 in the text. This figure has 6
vertices, 12 edges, and 8 faces.
164

• Eulers Polyhedron Formula
• An interesting thing occurs when working with
  polyhedron.

Think about what this means. Look at Figure 9.39
(c) on page 463 in the text. This figure has 6
vertices, 12 edges, and 8 faces.
165

• Polyhedrons
• A regular polyhedron is a polyhedron whose faces
  are all regular polygons of the same size and
  shape (i.e., a cube).
• A prism is a special polyhedron whose bases are
  congruent polygons and whose sides are
  parallelograms. The parallelogram regions are
  called the lateral faces of the prism.
• A right prism is a prism whose lateral faces are
  all rectangles.
• A few prisms are illustrated in Figure 9.40 page
  464 in the text.

166

• Volume Prism
• The volume of any prism can be found by
  multiplying the area of the base, B, by the
  height, h, of the prism.
• V Bh

167

• Volume Prism
• The volume of any prism can be found by
  multiplying the area of the base, B, by the
  height, h, of the prism.
• V Bh
• Study Example 4 on page 464 in the text. This
  example illustrates how to find the volume of a
  trapezoidal prism.

168

• Volume of Prism Example
• Find the volume of the prism shown.

169

• Volume of Prism Example
• Find the volume of the prism shown.

The first thing you have to ask yourself is,
What shape is the base?
170

• Volume of Prism Example
• Find the volume of the prism shown.

The prism is sitting on a rectangle, but the base
is actually a triangle. Why? In a prism, there
is a top base and a bottom base, and the two
bases are parallel and congruent. The rectangle
cannot be the base, since there is not another
rectangular face parallel to it.
171

• Volume of Prism Example
• Find the volume of the prism shown.

The formula for the volume of a prism is V
Bh B area of base B ½(bh) ½(10x6) 30 So,
V 30x3 90 ft3
172

• Volume Pyramids
• A pyramid is a polyhedron which has only one
  base. Pyramids are named by their base. If a
  pyramid has a triangle for its base, it is called
  a triangular pyramid.
• Figure 9.43 on page 465 in the text illustrates
  four different pyramids. Notice how all of the
  faces except for one (the base) are triangles.
• Figure 9.44 on page 466 in the text shows a
  pyramid inside of a prism. Notice how the volume
  of the pyramid is less than the volume of the
  prism.

173

• Volume Pyramids
• It turns out that it would take three pyramids to
  equal the volume of the prism with the same base
  and height.
• Volume of a pyramid

V 1/3(Bh)
174

• Volume Pyramids
• It turns out that it would take three pyramids to
  equal the volume of the prism with the same base
  and height.
• Volume of a pyramid
• Study Example 6 on page 466 in the text. This
  example illustrates how to find the volume of a
  pyramid with a square base.

V 1/3(Bh)
175

• Volume of Pyramid Example
• Look at the diagram for problem 18 on page 467 in
  the text. We are asked to find the volume of
  this pyramid.

176

• Volume of Pyramid Example
• Look at the diagram for problem 18 on page 467 in
  the text. We are asked to find the volume of
  this pyramid.
• Remember that the formula for a pyramid is
• V 1/3(Bh). Since the base is a triangle, the
  area of the triangular base is B ½(bh).
• B ½(bh) ½(9x15) ½(135) 67.5 B

177

• Volume of Pyramid Example
• Look at the diagram for problem 18 on page 467 in
  the text. We are asked to find the volume of
  this pyramid.
• Remember that the formula for a pyramid is
• V 1/3(Bh). Since the base is a triangle, the
  area of the triangular base is B ½(bh).
• B ½(bh) ½(9x15) ½(135) 67.5 B
• V 1/3(Bh) 1/3(67.5x13) 1/3(877.5) 292.5
  ft3
• The volume of the pyramid is 292.5 ft3.

178

• Volume Homework Examples
• Look at the diagram for problem 24 on page 468 in
  the text. We are asked to find the volume of the
  shaded area.

179

• Volume Homework Examples
• Look at the diagram for problem 24 pm page 468 in
  the text. We are asked to find the volume of the
  shaded area.
• This is a problem that can be solved using
  subtraction. Find the volume of the cube, find
  the volume of the sphere, and then subtract the
  two answers. To find the volume of the sphere,
  you need to know the radius of the sphere. Since
  the sphere is 4 feet across, the radius of the
  sphere must be 2 feet.
• Try to solve this problem. The solution follows.

180

• Volume Homework Examples
• The volume of the cube is V s3 43 64 ft3.

181

• Volume Homework Examples
• The volume of the cube is V s3 43 64 ft3.
• The volume of the cylinder is
• V 4/3(?r3) 4/3(3.14 x 23) 4/3( 3.14 x 8)
• 4/3 ( 25.12 ) 33.49 ft3.

182

• Volume Homework Examples
• The volume of the cube is V s3 43 64 ft3.
• The volume of the cylinder is
• V 4/3(?r3) 4/3(3.14 x 23) 4/3( 3.14 x 8)
• 4/3 ( 25.12 ) 33.49 ft3.
• The volume of the shaded area is found by
  subtracting the volume of the sphere from the
  volume of the cube.

183

• Volume Homework Examples
• The volume of the cube is V s3 43 64 ft3.
• The volume of the sphere is
• V 4/3(?r3) 4/3(3.14 x 23) 4/3( 3.14 x 8)
• 4/3 ( 25.12 ) 33.49 ft3.
• Volume of the shaded area is found by subtracting
  the volume of the sphere from the volume of the
  cube.
• V shaded area 64 33.49 30.51 ft3.

184

• 9.5 - The Möbius Strip, Klein Bottle, and Maps
• This section covers a branch of mathematics
  called topology. Topology is sometimes referred
  to as rubber sheet geometry. Read on and see
  if you can figure out why.

185

• The Möbius Strip
• August Ferdinand Möbius is best known for his
  studies of the properties of one-sided surfaces,
  one of which is called a Möbius strip.
• A Möbius strip, also called a Möbius band, is a
  one-sided, one-edged surface.
• Follow the instructions on page 471 in the text
  to make a Möbius strip. Work through the four
  Experiments on the same page to learn some
  interesting properties of the Möbius strip.

186

• The Möbius Strip
• In Experiment 1, on page 471 in the text, you
  should have discovered that a Möbius strip has
  one edge.
• In Experiment 2, on page 471 in the text, you
  should have discovered that a Möbius has one
  surface.
• In Experiment 3, on page 471 in the text, you
  should have created one long loop (one strip).
• In Experiment 4, on page 471 in the text, you
  should have created a long loop and a shorter
  loop (two strips).

187

• The Klein Bottle
• The punctured Klein bottle resembles a bottle but
  only has one side. It was named after Felix
  Klein.
• Look closely at the model of the Klein bottle
  shown in Figure 9.52 on page 472 in the text.
  The Klein bottle has only one edge and no outside
  or inside because it has just one side.
• Read the second paragraph on page 472 in the
  text. It starts with, Imagine trying to paint a
  Klein bottle.

188

• Maps
• Maps have interested topologists for years. Map
  makers have known for a long time that no matter
  whether a map is drawn on a flat surface or a
  sphere, and no matter how complex the diagram is,
  only four colors are needed to differentiate each
  country (or state) from its immediate neighbor.

189

• Maps
• Maps have interested topologists for years. Map
  makers have known for a long time that no matter
  whether a map is drawn on a flat surface or a
  sphere, and no matter how complex the diagram is,
  only four colors are needed to differentiate each
  country (or state) from its immediate neighbor.
• Meaning, every map can be drawn by using only
  four colors, and no two countries with a common
  border will have the same color. Note
  countries which meet in a single point are not
  considered to have a common border.

190

• Maps
• Originally no one had proved that you only needed
  four colors, but it was accepted as being the
  case as early as 1852. Then, in 1976 two people
  from the University of Illinois proved the
  four-color problem using a computer.

191

• Maps
• Originally no one had proved that you only needed
  four colors, but it was accepted as being the
  case as early as 1852. Then, in 1976 two people
  from the University of Illinois proved the
  four-color problem using a computer.
• Read page 473 in the text for a brief description
  on how they went about solving the four-color
  problem. When you read this, keep in mind what
  computers were like in 1976, definitely not what
  we have today.

192

• Jordan Curves
• A Jordan curve is a topological object that can
  be thought of as a circle twisted out of shape.
  Look at Figure 9.57 on page 474 in the text.
  This shows the formation of a Jordan curve. Take
  a circle, deform it, and spiral it up.
• Since a Jordan curve is a circle, it has an
  inside and an outside. With a circle, it is easy
  to tell if a point is inside the circle or
  outside the circle. But how can you tell if a
  point is inside or outside of a Jordan curve?

193

• Jordan Curves
• A quick way to tell whether a point is inside or
  outside the curve is to draw a straight line from
  the point to a point that is clearly outside the
  curve. If the straight line crosses the curve in
  an even number of points, the point is outside.
  If the straight line crosses the curve in an odd
  number of points, the point is inside the curve.

194

• Jordan Curves
• A quick way to tell whether a point is inside or
  outside the curve is to draw a straight line from
  the point to a point that is clearly outside the
  curve. If the straight line crosses the curve in
  an even number of points, the point is outside.
  If the straight line crosses the curve in an odd
  number of points, the point is inside the curve.
• Do you know why this works? Can you explain it
  in words?

195

• Jordan Curves
• Here is a Jordan curve. Determine if the point
  is inside, or outside of the curve.

196

• Jordan Curves
• Here is a Jordan curve. Determine if the point
  is inside, or outside of the curve.

Draw a straight line from the point to a point
that is clearly outside the curve.
197

• Jordan Curves
• Here is a Jordan curve. Determine if the point
  is inside, or outside of the curve.

Draw a straight line from the point to a point
that is clearly outside the curve.
198

• Jordan Curves
• Here is a Jordan curve. Determine if the point
  is inside, or outside of the curve.

The straight line crosses the curve four times.
Thus, the point is outside of the curve.
199

• Jordan Curves
• Here is a Jordan curve. Determine if the point
  is inside, or outside of the curve.

Of course it is much easier to see it is outside
the curve if you can shade the curve.

200

• Topological Equivalence
• A doughnut equals a coffee cup?

201

• Topological Equivalence
• A doughnut equals a coffee cup?
• Two geometric figures are said to
  be topologically equivalent if one figure can
  be elastically twisted, stretched, bent, or
  shrunk into the other figure without puncturing
  or ripping the original figure.

202

• Topological Equivalence
• A doughnut equals a coffee cup?
• Two geometric figures are said to
  be topologically equivalent if one figure can
  be elastically twisted, stretched, bent, or
  shrunk into the other figure without puncturing
  or ripping the original figure.
• Look at Figure 9.58 on page 474 in the text.
  This shows how a doughnut is topologically
  equivalent to a coffee cup. Not something you
  would normally think about unless you were a
  topologist!

203

• Topological Equivalence
• In topology, figures are classified according to
  their genus. The genus of an object is
  determined by the number of holes in the object.
  If two objects have the same genus, then they are
  topologically equivalent.

204

• Topological Equivalence
• In topology, figures are classified according to
  their genus. The genus of an object is
  determined by the number of holes in the object.
  If two objects have the same genus, then they are
  topologically equivalent.
• So, the doughnut had one hole, and the coffee cup
  had one hole (the handle), so they were
  topologically equivalent.

205

• Topological Equivalence
• In topology, figures are classified according to
  their genus. The genus of an object is
  determined by the number of holes in the object.
  If two objects have the same genus, then they are
  topologically equivalent.
• So, the doughnut had one hole, and the coffee cup
  had one hole (the handle), so they were
  topologically equivalent.
• See Figure 9.59 on page 474 in the text to see
  more objects that are topologically equivalent
  (Note read down the columns of the chart).

206

• 9.6 - Non-Euclidean Geometry and Fractal Geometry
• The beginning of this chapter introduced plane
  geometry. We accepted the postulates and axioms
  of plane geometry as true. This section
  discusses what happens when you do not accept
  these postulates and axioms as true.

207

• The Euclidean Parallel Postulate
• Plane geometry is based on Euclids fifth
  postulate.
• The fifth postulate stated, If a straight line
  falling on two straight lines makes the interior
  angles on the same side less than two right
  angles, the two straight lines, if produced
  indefinitely, meet on that side on which the
  angles are less than the two right angles.
• A diagram may help clarify what this postulate
  means.

208

• The Euclidean Parallel Postulate

The red line is the line falling on two straight
lines.
A
B
209

• The Euclidean Parallel Postulate

The interior angles on the same side are ? A
? B.
A
B
210

• The Euclidean Parallel Postulate

If ? A ? B are less than two right angles, then
the lines, if continued, will intersect.
Lines intersect here
A
B
211

• The Euclidean Parallel Postulate

Notice that the lines intersect on the side of
the red line where the angles are less than two
right angles.
A
B
212

• The Euclidean Parallel Postulate
• In 1795, John Playfair, a Scottish physicist and
  mathematician wrote a geometry book. In this
  book, he gave a logically equivalent
  interpretation of Euclids fifth postulate. This
  version is often referred to as Playfairs
  postulate or the Euclidean parallel postulate.
  This postulate is a little easier to understand.
• Euclidean parallel postulate Given a line and a
  point not on the line, one and only one line can
  be drawn through the given point parallel to the
  given line.

213

• The Euclidean Parallel Postulate
• Given a line and a point not on the line, one and
  only one line can be drawn through the given
  point parallel to the given line.

214

• The Euclidean Parallel Postulate
• Given a line and a point not on the line, one and
  o