Geometry and Measurement

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Geometry and Measurement
Chapter Nine
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Lines and Angles
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UNIT 20

• ANGULAR MEASURE

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A plane is a flat surface that extends
indefinitely.
Plane
Space extends in all directions indefinitely.
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Point
The most basic concept of geometry is the idea of
a point in space. a point has no length, no
width, and no height, but it does have location.
We will represent a point by a dot, and we will
label points with letters.
A
Point A
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(No Transcript)
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An angle is made up of two rays that share the
same endpoint called a vertex.
Vertex
The angle can be named ?ABC, ?CBA, ?B or ?x.
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An angle can be measured in degrees. There are
360º (degrees) in a full revolution or full
circle.
360º
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Classifying Angles
Angle Measure
Examples
Name
Acute Angle
Between 0 and 90
Right Angle
Exactly 90
Obtuse Angle
Between 90 and 180
Straight Angle
Exactly 180º
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Two angles that have a sum of 90 are called
complementary angles.
Two angles that have a sum of 180 are called
supplementary angles.
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Two lines in a plane can be either parallel or
intersecting. Parallel lines never meet.
Intersecting lines meet at a point. The symbol
?? is used to denote is parallel to.
Parallel lines
Intersecting lines
p ?? q
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Two lines are perpendicular if they form right
angles when they intersect. The symbol ? is
used to denote is perpendicular to.
n ? m
Perpendicular lines
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When two lines intersect, four angles are formed.
Two of these angles that are opposite each other
are called vertical angles. Vertical angles have
the same measure.
?a ?c
?d ?b
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Two angles that share a common side are called
adjacent angles. Adjacent angles formed by
intersecting lines are supplementary. That is,
they have a sum of 180 .
?a and ?b
?b and ?c
?c and ?d
?d and ?a
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A line that intersects two or more lines at
different points is called a transversal.
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Parallel Lines Cut by a Transversal

• If two parallel lines are cut by a transversal,
  then the measures of corresponding angles are
  equal and alternate interior angles are equal.

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Corresponding angles are equal.
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Alternate interior angles are angles on opposite
sides of the transversal between the two parallel
lines.
a
b
c
d
e
f
g
h
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What is the sum of the measures of the interior
angles of a triangle?
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B
A
C
The sum of the measures of the angles of any
triangle is 180.
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A right triangle is a triangle in which one of
the angles is a right angle or measures 90º
(degrees).
The hypotenuse of a right triangle is the side
opposite the right angle. The legs of a right
triangle are the other two sides.
hypotenuse
leg
leg
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Helpful Hint

• The sum of the two acute angles in a right
  triangle is 90?.

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TERMS DEFINED

• An angle is a figure made by two lines that
  intersect
• An angle is also described as the union of two
  rays having a common end point
• The two rays are called the sides of the angle,
  and their common end point is called a vertex
• Angles are measured in degrees. The degree symbol
  is the angle symbol is ?
• The degree of precision required in measuring and
  computing angles depends on how the angle is used
• A complete circle equals 360

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UNITS OF ANGULAR MEASURE

• The decimal degree is generally the preferred
  unit of measurement in metric calculations
• In the United States it is customary to express
  angular measure in the following ways
• As decimal degrees, such as 9.7 degrees
• As fractional degrees, such as 36 1/2 degrees
• As degrees, minutes, and seconds, such as 63
  degrees, 27 minutes, 48 seconds
• A degree is divided in 60 equal parts called
  minutes (?)
• A minute is divided in 60 equal parts called
  seconds (?)

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EXPRESSING DEGREES, MINUTES, AND SECONDS AS
DECIMAL DEGREES

• Procedure for expressing degrees, minutes, and
  seconds as decimal degrees
• Divide the seconds by 60 to obtain the decimal
  minute
• Add the decimal minute to the given number of
  minutes
• Divide the sum of minutes by 60 to obtain the
  decimal degrees
• Add the decimal degree to the given number of
  degrees

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CONVERSION EXAMPLE

• Express 7357?48? as decimal degrees
• Divide 48 seconds by 60 to obtain decimal minutes
• Add the decimal minutes to the given minutes
• Divide sum of minutes by 60 to obtain decimal
  degrees
• Add the decimal degree to the given degree

48? ? 60 0.8 minutes
57? 0.8? 57.8?
57.8? ? 60 0.9633? (Rounded)
73 0.9633 73.9633 Ans
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EXPRESSING DECIMAL DEGREES AS DEGREES, MINUTES,
AND SECONDS

• Procedure for expressing decimal degrees as
  degrees, minutes, and seconds
• Multiply the decimal part of the degrees by 60
  minutes to obtain minutes
• If the number of minutes obtained is not a whole
  number, multiply the decimal part of the minutes
  by 60 seconds to obtain seconds
• Combine degrees, minutes, and seconds

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CONVERSION EXAMPLE

• Express 48.54? as degrees, minutes, and seconds
• Multiply the decimal part of the degrees by 60 to
  obtain minutes
• Multiply the decimal part of the minutes by 60 to
  obtain seconds
• Combine degrees, minutes, and seconds

.54 60 32.4 minutes
.4 60 24 seconds
48.54 48 32? 24? Ans
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ADDING ANGLES

• Add the following angles
• 49 53? 37?
• 38? 47? 24? Add seconds to seconds,
  minutes to minutes, etc.
• Simplify the sum

87 100? 61?
61? 1? 1?

• Now add the 1? to the 100? or 100? 1? 101?

• Change 101? to degrees 101? 1? 41?

• Add the 1 degree to the 87 degrees and combine
  all the units

87? 1? 88?, so we end up with 88? 41? 1? Ans
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SUBTRACTING ANGLES

• Subtract the following angles 89? 23? 15?
  70? 35? 20?
• 20? cannot be subtracted from 15?, so borrow 60?
  from the 23?
• Also, 35? cannot be subtracted from the 22? that
  were left after we borrowed, so we will need to
  borrow 60? from the 89?
• We can now complete our subtraction

88? 82? 75? 70? 35? 20? 18? 47? 55?
Ans
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MULTIPLYING ANGLES

• Multiply 51? 33? 42? by 351? 33? 42?
• ? 3
• Simplify the product

153 99? 126?
126? 2? 6?

• Adding these 2? to the 99?, we now have 101?
  101? 1? 41?

• Adding this degree to the 153 degrees, we now
  have 154

• Combining units, we now have 154? 41? 6? Ans

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DIVIDING ANGLES

• Divide 147? 55? 34? by 2

73?
57?
47? Ans
146?
1 60?
55?
115?
114?
1? 60?
34?
94?
94?
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COMPLEMENTS AND SUPPLEMENTS

• Two angles are complementary when their sum is
  90
• Two angles are supplementary when their sum is
  180?
• Determine the complement of 43 18?
• Subtract 43? 18? from 90? (or 89? 60?) to find
  its complement

89? 60? 43? 18? 46? 42? Ans
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PRACTICE PROBLEMS

• Express the following degrees, minutes, and
  seconds as decimal degrees. Round to three
  decimal places where necessary
• 1. 143? 54? 32?
• 2. 242? 33? 24?
• Express the following decimal degrees as degrees,
  minutes, and seconds
• 3. 129.76
• 4. 85.845
• Perform the following operations. Simplify all
  answers
• 5. 45 54? 39? 79? 17? 43?
• 6. 87? 16? 25? - 76? 21? 36?

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PRACTICE PROBLEMS

• 7. 54? 47? 32? 16? 19? 35?
• 8. 72? 15? 15? - 60? 20? 20?
• 9. 43? 33? 29? ? 3
• 10. 54? 48? 15? ? 3
• 11. 136? 58? 45? ? 4
• 12. 272? 38? 52? ? 4
• 13. Determine the complement of 49?15? 16?
• 14. Determine the supplement of 49? 15? 16?
• 15. Determine the supplement of 147? 36? 21?

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PROBLEM ANSWER KEY

• 1. 143.909 12.
  68? 9? 43?
• 2. 242.557 13.
  40? 44? 44?
• 3. 129? 45? 36? 14.
  130? 44? 44?
• 4. 85? 50? 42? 15.
  32 23? 39?
• 5. 125? 12? 22?
• 6. 10? 54? 49?
• 7. 71? 7? 7?
• 8. 11? 54? 55?
• 9. 130? 40? 27?
• 10. 18? 16? 5?
• 11. 547? 55? 0?