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MM150 Unit Six Seminar
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Professor DeLong
profsdelong (AIM name)
Copyright © 2009 Pearson Education, Inc.
Gosh, I love
geometry…
let’s get
started!!!
Slide 9 - 1
6.1
Points, Lines, Planes, and
Angles
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Slide 9 - 2
Basic Terms
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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.
A line segment is part of a line between two
points, including the endpoints.
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Slide 9 - 3
Basic Terms
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
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Angles
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An angle is the union of two rays with a
common endpoint; denoted R.
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:
RABC, RCBA, RB
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Slide 9 - 5
Angles
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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 Angle is greater than 90 but less
than 180
 Straight Angle is 180
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Slide 9 - 6
Types of Angles
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Adjacent Angles-angles that have a common
vertex and a common side but no common
interior points.
Complementary Angles-two angles whose sum
of their measures is 90 degrees.
Supplementary Angles-two angles whose sum
of their measures is 180 degrees.
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Slide 9 - 7
Example
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If RABC and RCBD 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
mRABC  154.2o
mRCBD  25.7o
Slide 9 - 8
More definitions
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Vertical angles are the nonadjacent angles
formed by two intersecting straight lines.
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 - 9
6.2
Polygons
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Slide 9 - 10
Polygons
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Polygons are named according to their number
of sides.
Number of
Sides
3
Name
Triangle
Number of
Sides
8
Octagon
4
Quadrilateral
9
Nonagon
5
Pentagon
10
Decagon
6
Hexagon
12
Dodecagon
7
Heptagon
20
Icosagon
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Name
Slide 9 - 11
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The sum of the measures of the interior angles
of an n-sided polygon is
(n  2)180.
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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.
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Slide 9 - 12
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Determine the sum of
the interior angles.
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1080
 135
8
S  (n  2)180
 (8  2)(180 )
 6(180 )
 1080
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The measure of one
interior angle is
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The exterior angle is
supplementary to the
interior angle, so
180  135 = 45
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Types of Triangles
Acute Triangle
All angles are acute.
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Obtuse Triangle
One angle is obtuse.
Slide 9 - 14
Types of Triangles continued
Right Triangle
One angle is a right
angle.
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Isosceles Triangle
Two equal sides.
Two equal angles.
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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 - 16
Similar Figures
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Two polygons are similar if their corresponding
angles have the same measure and the lengths
of their corresponding sides are in proportion.
9
6
4
4
3
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6
6
4.5
Slide 9 - 17
Example
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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
5
75
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12
Slide 9 - 18
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 - 19
Congruent Figures
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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 - 20
Quadrilaterals
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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 - 21
Classifications
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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 - 22
Classifications continued
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Rhombus
Both pairs of opposite
sides are parallel.
The four sides are
equal in length.
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Rectangle
Both pairs of opposite
sides are parallel.
Both pairs of opposite
sides are equal in
length. The angles
are right angles.
Slide 9 - 23
Classifications continued
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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 - 24
6.3
Perimeter and Area
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Slide 9 - 25
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 = s1 + s2 + b1 + b2 A  21 h(b1  b2 )
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Example
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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.
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Slide 9 - 27
Example continued
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a) The area of the roof is
A = lw
A = 30(50)
A = 1500 ft2
1500  2 (both sides of the roof) = 3000 ft2
b) Determine the number of squares
area of roof
3000

 30
area of one square 100
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Slide 9 - 28
Example continued
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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 - 29
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
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b
Slide 9 - 30
Example
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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.
38 ft rope
12 ft
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Slide 9 - 31
Example continued
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a2  b2  c 2
122  b 2  38 2
144  b 2  1444
b 2  1300
b 2  1300
b  36.06
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12
38
b
The distance is approximately 36.06 feet.
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Slide 9 - 32
Circles
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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.
A diameter, d, of a circle
is a line segment through
the center of the circle with
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 - 33
Example
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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 - 34
6.4
Volume and Surface Area
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Slide 9 - 35
Volume
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Volume is the measure of the capacity of a
figure.
It is the amount of material you can put inside a
three-dimensional figure.
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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 - 36
Volume Formulas
Figure
Rectangular
Solid
Cube
Formula
V = lwh
Cylinder
V = r2h
Cone
Sphere
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Diagram
V = s3
V  31  r 2 h
V  r
4
3
3
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Surface Area Formulas
Figure
Formula
Rectangular SA = 2lw + 2wh +2lh
Solid
Cube
SA= 6s2
SA = 2rh + 2r2
Cylinder
Cone
Diagram
SA   r
Sphere
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2
 r
r
2
SA  4 r
h
2
2
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Example
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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? How many
cubic feet is this?
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Slide 9 - 39
Example continued
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We need to find the volume of one planter.
V  lwh
V  12(8)(3)
V  288 in.3
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Soil for 500 planters would be
500(288) = 144,000 cubic inches
144,000

 83.33 ft 3
1728
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Slide 9 - 40
Polyhedron
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A polyhedron is a closed surface formed by the
union of polygonal regions.
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Slide 9 - 41
Euler’s Polyhedron Formula
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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  # of edges + # of faces = 2
x  12  6  2
x 6  2
x 8
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There are 8 vertices.
Slide 9 - 42
Volume of a Prism
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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.
Find the volume.
A  21 bh
A  21 (6)(4)
V  Bh
V  12(8)
4m
A  12 m2
8m
V  96 m3
6m
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Slide 9 - 43
Volume of a Pyramid
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V  31 Bh where B is the area of the base and h
is the height.
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Example: Find the volume of the pyramid.
Base area = 122 = 144
V  31 Bh
V  31 (144)(18)
18 m
V  864 m3
12 m
12 m
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Slide 9 - 44
If I need help, I
won’t lay an
egg, I’ll contact
Professor
DeLong!
Copyright © 2009 Pearson Education, Inc.
Slide 9 - 45