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SEVENTH EDITION and EXPANDED SEVENTH EDITION
Slide 9-1
Chapter 9
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Points, lines, planes, and angles
Polygons, similar figures, and congruent figures
Perimeter and area
Pythagorean theorem
Circles
Volume
Geometry
Copyright © 2005 Pearson Education, Inc.
9.1
Points, Lines, Planes, and
Angles
Copyright © 2005 Pearson Education, Inc.
Basic Terms

A point, line, and plane are three basic terms in
geometry that are NOT given a formal definition, yet
we recognize them when we see them.

A line is a set of points.

Any two distinct points determine a unique line.

Any point on a line separates the line into three
parts: the point and two half lines.

A ray is a half line including the endpoint.
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Slide 9-4
Basic Terms

A line segment is part of a line between two
points, including the endpoints.
Description
Diagram
Line AB
Ray AB
A
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B
B
A
A
AB
AB
B
A
Ray BA
Line segment AB
Symbol
B
BA
AB
Slide 9-5
Plane





We can think of a plane as a two-dimensional
surface that extends infinitely in both directions.
Any three points that are not on the same line
(noncollinear points) determine a unique plane.
A line in a plane divides the plane into three
parts, the line and two half planes.
Any line and a point not on the line determine a
unique plane.
The intersection of two distinct, non-parellel
planes is a line.
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Slide 9-6
Angles




An angle is the union of two rays with a
common endpoint; denoted .
The vertex is the point common to both rays.
The sides are the rays that make the angle.
There are several ways to name an angle:
ABC,
CBA,
B
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Slide 9-7
Angles



The measure of an angle is the amount of rotation
from its initial to its terminal side.
Angles can be measured in degrees, radians, or,
gradients.
Angles are classified by their degree measurement.




Right Angle is 90
Acute Angle is less than 90
Obtuse Angles is greater than 90 but less than 180 
Straight Angle is 180
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Slide 9-8
Types of Angles



Adjacent Angles-angles that have a common
vertex and a common side but no common
interior points.
Complementary Angles-two angles whose sum
is 90 degrees.
Supplementary Angles-two angles whose sum
is 180 degrees.
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Slide 9-9
Example

If ABC and ABD are supplementary and the
measure of ABC is 6 times larger than CBD,
determine the measure of each angle.
C
m ABC  m CBD  180
6 x  x  180
7 x  180
x  25.7
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A
B
D
m ABC » 154.3
m CBD » 25.7
Slide 9-10
More definitions



Vertical angles have the same measure.
A line that intersects two different lines, at two
different points is called a transversal.
Special angles are given to the angles formed
by a transversal crossing two parallel lines.
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Slide 9-11
Special Names
Alternate interior
angles
Alternate exterior
angles
Corresponding
angles
Interior angles on the
opposite side of the
transversal—have the
same measure
Exterior angles on the
opposite sides of the
transversal—have the
same measure
One interior and one
exterior angles on the
same side of the
transversal-have the same
measure
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1 2
3 4
5 6
7 8
1 2
3 4
5 6
7 8
1 2
3 4
5 6
7 8
Slide 9-12
9.2
Polygons
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Polygons

Polygons are names according to their number of sides.
Number of
Sides
Name
Number of
Sides
Name
3
Triangle
8
Octagon
4
Quadrilateral
9
Nonagon
5
Pentagon
10
Decagon
6
Hexagon
12
Dodecagon
7
Heptagon
20
Icosagon
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Slide 9-14
Triangles

The sum of the measures of the interior angles
of an n-sided polygon is (n  2)180.

Example: A certain brick paver is in the shape of
a regular octagon. Determine the measure of an
interior angle and the measure of one exterior
angle. The exterior
angle is supplementary to the interior
angle
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Slide 9-15
Triangles continued

Determine the sum of the
interior angles.

S  (n  2)180
 (8  2)(180 )
 6(180 )
 1080

The measure of one
interior angle is
1080
 135
8
The exterior angle is
supplementary to the
interior angle, so the
measure of one
exterior angle is
180  135 = 45
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Slide 9-16
Types of Triangles
Acute Triangle
All angles are acute.
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Obtuse Triangle
One angle is obtuse.
Slide 9-17
Types of Triangles (continued)
Right Triangle
One angle is a right
angle.
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Isosceles Triangle
Two equal sides.
Two equal angles.
Slide 9-18
Types of Triangles (continued)
Equilateral Triangle
Three equal sides.
Three equal angles,
60º each.
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Scalene Triangle
No two sides are
equal in length.
Slide 9-19
Similar Figures

Two polygons are similar if their corresponding
angles have the same measure and their
corresponding sides are in proportion.
9
6
4
4
3
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6
6
4.5
Slide 9-20
Example

Catherine Johnson wants to measure the height
of a lighthouse. Catherine is 5 feet tall and
determines that when her shadow is 12 feet
long, the shadow of the lighthouse is 75 feet
long. How tall is the lighthouse?
x
x
75
5
75
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12
Slide 9-21
Example continued
ht. lighthouse lighthouse's shadow
=
ht. Catherine Catherine's shadow
x 75

5 12
12x  375
x
x  31.25
5
75
12
Therefore, the lighthouse is 31.25 feet tall.
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Slide 9-22
Congruent Figures


If corresponding sides of two similar figures are
the same length, the figures are congruent.
Corresponding angles of congruent figures have
the same measure.
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Slide 9-23
Quadrilaterals


Quadrilaterals are four-sided polygons, the sum
of whose interior angles is 360.
Quadrilaterals may be classified according to
their characteristics.
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Slide 9-24
Classifications


Trapezoid
Two sides are parallel.
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
Parallelogram

Both pairs of opposite
sides are parallel. Both
pairs of opposite sides
are equal in length.
Slide 9-25
Classifications continued

Rhombus

Rectangle

Both pairs of opposite sides
are parallel. The four sides are
equal in length.

Both pairs of opposite sides
are parallel. Both pairs of
opposite sides are equal in
length. The angles are right
angles.
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Slide 9-26
Classifications continued

Square

Both pairs of opposite sides
are parallel. The four sides are
equal in length. The angles
are right angles.
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Slide 9-27
9.3
Perimeter and Area
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Formulas
Figure
Rectangle
Square
Parallelogram
Triangle
Trapezoid
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Perimeter
Area
P = 2l + 2w
A = lw
P = 4s
A = s2
P = 2b + 2w
A = bh
P = s1 + s2 + s3
A  21 bh
P = s 1 + s 2 + b 1 + b2
A  21 h(b1  b2 )
Slide 9-29
Example




Marcus Sanderson needs to put a new roof on his
barn. One square of roofing covers 100 ft2 and costs
$32.00 per square. If one side of the barn roof
measures 50 feet by 30 feet, determine
a) the area of the entire roof.
b) how many squares of roofing he needs.
c) the cost of putting on the roof.
side 2
side 1
Roof
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Slide 9-30
Example continued

a) The area of the roof is


A = lw
A = 30 ft  50 ft
A = 1500 ft2
1500 ft2
A = Both sides of the roof =
1500 ft2  2 = 3000 ft2
b) Determine the number of squares
area of roof
3000sq. ft.
=
= 30
area of one square 100sq. ft.
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Slide 9-31
Example continued

c) Determine the cost

30 squares  $32 per square
$960

It will cost a total of $960 to roof the barn.

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Slide 9-32
Pythagorean Theorem
The sum of the squares of the lengths of the
legs of a right triangle equals the square of the
length of the hypotenuse.
leg2 + leg2 = hypotenuse2
Symbolically, if a and b represent the lengths of
the legs and c represents the length of the
hypotenuse (the side opposite the right angle),
then
c
2
2
2
a
a +b =c
b
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Slide 9-33
Example

Tomas is bringing his boat into a dock that is 12 feet
above the water level. If a 38 foot rope is attached to the
dock on one side and to the boat on the other side,
determine the horizontal distance from the dock to the
boat.
12 ft
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38 ft rope
Slide 9-34
Example continued

a2 + b2 = c 2
122 + b 2 = 382
144 + b 2 = 1444
2
b = 1300
b = 1300
b » 36.06

38
12
b
The distance is approximately 36.06 feet.
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Slide 9-35
Circles




A circle is a set of points 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.
r
A diameter, d, of a circle
is a line segment through
d
the center of the circle with circumference
both end points on the circle.
The circumference is the length of the simple
closed curve that forms the circle.
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Slide 9-36
Example

Terri is installing a new circular swimming
pool in her backyard. The pool has a diameter
of 27 feet. How much area will the pool take
up in her yard? (Use π = 3.14.)
A  r
2
A   (13.5) The radius of the pool is 13.5 ft.
A  572.265 The pool will take up about 572
2
square feet.
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Slide 9-37
9.4
Volume
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Volume

Volume is the measure of the capacity of a
figure.
It is the amount of material you can put inside
a three-dimensional figure.

Surface area is sum of the areas of the
surfaces of a three-dimensional figure.
It refers to the total area that is on the outside
surface of the figure.
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Slide 9-39
Volume Formulas
Figure
Rectangular
Solid
Cube
Cylinder
Cone
Sphere
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Formula
V = lwh
Diagram
h
l
w
s
V = s3
s
V = r2h
s
r
h
V  31  r 2 h
V  r
4
3
3
h
r
Slide 9-40
Surface Area Formulas
Figure
Formula
Rectangular SA = 2lw + 2wh +2lh
Solid
Cube
SA= 6s2
Diagram
h
l
w
s
s
SA = 2rh + 2r2
Cylinder
Cone
r
s
h
r
SA   r
Sphere
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2
 r
r
2
SA  4 r
h
2
2
h
r
Slide 9-41
Example

Mr. Stoller needs to order potting soil for his
horticulture class. The class is going to plant
seeds in rectangular planters that are 12 inches
long, 8 inches wide and 3 inches deep. If the
class is going to fill 500 planters, how many
cubic inches of soil are needed?
3
8
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Slide 9-42
Example (continued)

We need to find the volume of one planter.
V  lwh
V  12(8)(3)
V  288 in.3


Soil for 500 planters would be
500(288) = 144,000 cubic inches
The number of cube feet
333
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Slide 9-43
Polyhedron

A polyhedron is a closed surface formed by the
union of polygonal regions.
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Slide 9-44
Euler’s Polyhedron Formula

Number of vertices  number of edges + number of
faces = 2

Example: A certain polyhedron has 12 edges and 6
faces. Determine the number of vertices on this
polyhedron.

# of vertices  # edges + # faces = 2
x  12  6  2
There are 8 vertices.
x 6  2
x 8
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Slide 9-45
Volume of a Prism



A prism is a polyhedron whose bases are
congruent and whose sides are
parallelograms.
V = Bh, where B is the area of the base and h
is the height.
Example: Find the volume of the figure.

Area of one triangle.
A  21 bh
A  21 (6)(4)
Find the volume.
V  Bh
V  12(8)
4m
A  12 m2
8m
V  96 m3
6m
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Slide 9-46
Volume of a Pyramid



A pyramid is a polyhedron with one base, all of
whose faces intersect at a common vertex.
V  31 Bh where B is the area of the base and h
is the height.
Example: Find the volume of the pyramid.
Base area = 122 = 144
V  31 Bh
18 m
V  31 (144)(18)
V  864 m3
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12 m
12 m
Slide 9-47