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Chapter 6
Geometry
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WHAT WE WILL COVER
• Points, lines, planes, and angles
• Polygons, similar figures, and congruent
figures
• Perimeter and Area
• Pythagorean theorem
• Circles
• Volume and Surface Area
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6.1
Points, Lines, Planes, and Angles
Page 216
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Basic Terms
• A point has no dimension.
• A line, a set of points, has 1 dimension.
• A line can be determined by any two
distinct points.
• 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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Basic Terms
Description
Diagram
Line AB
Ray AB
A
B
B
A
A
AB
AB
B
A
Ray BA
Line segment AB
Symbol
B
BA
AB
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Plane
• A plane is a two-dimensional surface that
extends infinitely in both directions.
• Any three points that are not on the same
line 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 planes is a line.
Page 220
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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:
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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 90o
– Acute Angle is less than 90o
– Obtuse Angle is greater than 90o but less
than 180o
– Straight Angle is 180o
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Some 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 of their measures is 90 degrees.
• Supplementary Angles-two angles whose
sum of their measures is 180 degrees.
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Example
• If
are supplementary and
the measure of
is 6 times larger than
, determine the measure of each
angle.
C
A
B
D
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More definitions
• 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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Special Names
Alternate
interior angles
Alternate
exterior angles
Corresponding
angles
Page 224
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 angle on the
same side of the
transversal–have the
same measure
1
2
3
4
5 6
7 8
1
3
2
4
5 6
7 8
1
3
2
4
5 6
7 8
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Example
• Find the measure of the other angles
94o
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6.2
Polygons
Page 229
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Polygons
• Polygons are named according to their
number of sides.
Number of Name
Sides
3
Triangle
Number of
Sides
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Name
Octagon
4
Quadrilateral 9
Nonagon
5
Pentagon
10
Decagon
6
Hexagon
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Dodecagon
7
Heptagon
20
Icosagon
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• The sum of the measures of the interior
angles of an n-sided polygon is
(n - 2)180o.
• 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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• 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
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• The exterior angle is
supplementary to the
interior angle, so
180o - 135o = 45o
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Types of Triangles
Acute Triangle
All angles are acute.
Obtuse Triangle
One angle is obtuse.
Right Triangle
One angle is a right
angle.
Isosceles Triangle
Two equal sides.
Two equal angles.
Equilateral Triangle
Three equal sides. Three
equal angles (60º) each.
Scalene Triangle
No two sides are equal
in length.
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Similar Figures
• Two polygons are similar if their
corresponding angles have the same
measure and the lengths of their
corresponding sides are in proportion.
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4
4
3
6
6
4.5
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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
5
75
12
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Example continued
ht. lighthouse lighthouse's shadow
=
ht. Catherine Catherine's shadow
x 75

5 12
12x  375
x
x  31.25
5
75
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Therefore, the lighthouse is 31.25 feet tall.
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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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Quadrilaterals
• Quadrilaterals are four-sided polygons, the
sum of whose interior angles is 360o.
• Quadrilaterals may be classified according
to their characteristics.
•Trapezoid
•Parallelogram
•Rhombus
Page 234
•Rectangle
•Square
6.3
Perimeter and Area
Page 240
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Formulas
Figure
Perimeter
Area
Triangle
P = s1 + s2 + s3
A  21 bh
Square
P = 4s
A = s2
Parallelogram
P = 2b + 2w
A = bh
Rectangle
P = 2l + 2w
A = lw
Trapezoid
P = s1 + s2 + b1 + b2
A  21 h(b1  b2 )
Page 243
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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.
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Example continued
• a) The area of the roof is
A = lw
A = 30(50)
A = 1500 ft2
1500 x 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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Example continued
• c) Determine the cost
30 squares x $32 per square
$960
It will cost a total of $960 to roof the barn.
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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
a 2 + b 2 = c2
c
Page 244
b
a
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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.
38 ft rope
12 ft
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Example continued
•
a2  b2  c 2
122  b 2  38 2
144  b 2  1444
b 2  1300
b  1300
b  36.06
2
12
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b
• The distance is approximately 36.06 feet.
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Page 245
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.
• 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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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.)
The radius of the pool is
13.5 ft.
The pool will take up about
572 square feet.
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6.4
Volume and Surface Area
Page 252
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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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Volume Formulas
Figure
Rectangular
Solid
Cube
Formula
V = lwh
Cylinder
V = r2h
Cone
Sphere
Diagram
V = s3
V  31  r 2 h
V  r
4
3
3
Page 255
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Surface Area Formulas
Figure
Formula
Rectangular Solid
SA = 2lw + 2wh +2lh
Cube
Cylinder
Cone
Sphere
Diagram
SA= 6s2
SA = 2rh + 2r2
SA   r
2
 r
r
2
SA  4 r
h
2
2
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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? How many cubic feet is this?
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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
144,000

 83.33 ft 3
1728
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Polyhedron
• A polyhedron is a closed surface formed
by the union of polygonal regions.
Page 258
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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 - # of edges + # of faces = 2
x  12  6  2
x 6  2
x 8
There are 8 vertices.
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Volume of a Prism
• V = Bh, where B is the area of the base,
look up the correct formula, 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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Volume of a Pyramid
•
V  31 Bh
where B is the area of the base,
look up the correct formula, and h is the
height.
• 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
Page 261
12 m
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