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Geometry and MeasurementGeometry and Measurement
Chapter NineChapter Nine
Lines and AnglesLines and Angles
Section 9.1
Plane
A A planeplane is a flat surface that extends is a flat surface that extends indefinitely.indefinitely.
SpaceSpace extends in all directions extends in all directions indefinitely.indefinitely.
3Martin-Gay, Prealgebra, 5ed
PointPoint
The most basic concept of geometry is the The most basic concept of geometry is the idea of a idea of a pointpoint in space. A point has no in space. A point has no length, no width, and no height, but it length, no width, and no height, but it does have location. We will represent a does have location. We will represent a point by a dot, and we will label points point by a dot, and we will label points with letters.with letters.
A
Point A
4Martin-Gay, Prealgebra, 5ed
A B
A B
Line AB or AB
Ray AB or AB
Line Segment AB or AB
A B
5Martin-Gay, Prealgebra, 5ed
Vertex
AnAn angleangle is made up of two rays that share is made up of two rays that share the same endpoint called athe same endpoint called a vertexvertex..
The angle can be named The angle can be named ABC, , CBA, , B,, or or x. .
A
BC
x
The vertex is themiddle point.
6
An angle can be measured in degrees. An angle can be measured in degrees. There are 360º (degrees) in a full There are 360º (degrees) in a full revolution or full circle.revolution or full circle.
360º
7Martin-Gay, Prealgebra, 5ed
Classifying AnglesClassifying Angles
Acute Angle
NameName Angle MeasureAngle Measure ExamplesExamples
Between 0° and 90°
Right Angle
Exactly 90°
Obtuse Angle
Between 90° and 180°
Straight Angle
Exactly 180º8
Two angles that have a sum of 90° are Two angles that have a sum of 90° are calledcalled complementary anglescomplementary angles..
Two angles that have a sum of 180° are Two angles that have a sum of 180° are calledcalled supplementary anglessupplementary angles..
9Martin-Gay, Prealgebra, 5ed
Two lines in a plane can be either parallel or Two lines in a plane can be either parallel or intersecting.intersecting. Parallel linesParallel lines never meet.never meet. Intersecting linesIntersecting lines meet at a point. The symbol meet at a point. The symbol is used to denote “is parallel to.” is used to denote “is parallel to.”
p
q
Parallel lines Intersecting linesp q
10Martin-Gay, Prealgebra, 5ed
Two lines are Two lines are perpendicularperpendicular if they form right if they form right angles when they intersect. The symbol angles when they intersect. The symbol is is used to denote “is perpendicular to.”used to denote “is perpendicular to.”
Perpendicular lines
n
m
n m
11Martin-Gay, Prealgebra, 5ed
When two lines intersect, four angles are formed. When two lines intersect, four angles are formed. Two of these angles that are opposite each other Two of these angles that are opposite each other are called are called vertical anglesvertical angles. Vertical angles have the . Vertical angles have the same measure.same measure.
a
bc
d
a c d b
12Martin-Gay, Prealgebra, 5ed
Two angles that share a common side are Two angles that share a common side are called called adjacent anglesadjacent angles. Adjacent angles formed . Adjacent angles formed by intersecting lines are supplementary. That by intersecting lines are supplementary. That is, they have a sum of 180 °.is, they have a sum of 180 °.
a and b b and c
c and d d and a
a
bc
d
13Martin-Gay, Prealgebra, 5ed
A line that intersects two or more lines at A line that intersects two or more lines at different points is called adifferent points is called a transversaltransversal..
14Martin-Gay, Prealgebra, 5ed
Parallel Lines Cut by a TransversalParallel Lines Cut by a Transversal
If two parallel lines are cut by a If two parallel lines are cut by a transversal, then the measures of transversal, then the measures of corresponding angles are equalcorresponding angles are equal and and alternate interior angles are equalalternate interior angles are equal..
15Martin-Gay, Prealgebra, 5ed
Corresponding angles are equal.Corresponding angles are equal.
a e
a bc d
e fg h
b f
d h c g 16
Martin-Gay, Prealgebra, 5ed
Alternate interior anglesAlternate interior angles are angles on are angles on opposite sides of the transversal between opposite sides of the transversal between the two parallel lines.the two parallel lines.
c f
a bc d
e fg h
d e
17Martin-Gay, Prealgebra, 5ed
PerimeterPerimeter
Section 9.2
Two measures of plane figures are Two measures of plane figures are important:important:
• the distance around a plane figure the distance around a plane figure called the called the perimeterperimeter or or circumferencecircumference
• the number of square units in the the number of square units in the interior of a plane figure called the interior of a plane figure called the areaarea..
andand
19Martin-Gay, Prealgebra, 5ed
ll w w
P 2l 2w
Perimeter of a RectanglePerimeter of a Rectangle
Perimeter 2 • length 2 • width
width
length
width
length
20Martin-Gay, Prealgebra, 5ed
Perimeter is always measured in units.Perimeter is always measured in units.
Helpful HintHelpful Hint
21Martin-Gay, Prealgebra, 5ed
ss s s
P 4s
Perimeter of a SquarePerimeter of a Square
Perimeter 4 • side
sideside
side
side
22Martin-Gay, Prealgebra, 5ed
b
a c
P a b c
Perimeter of a TrianglePerimeter of a Triangle
The perimeter of every polygon may be The perimeter of every polygon may be found by adding all the sides.found by adding all the sides.
a b c
P side a side b side c
23
The The circumferencecircumference is the distance around is the distance around a circle.a circle.
Circumference diameter always results in the same ratio. This number is named “pi” and is approximately () equal to
or 3.147
22
diameter
24Martin-Gay, Prealgebra, 5ed
Circumference of a CircleCircumference of a Circle
r
d
Circumference 2··radius
Circumference ·diameter
C 2r oror C d
oror
25Martin-Gay, Prealgebra, 5ed
P a b c
P a b c d
P 2l 2w
P 4s
P a b c d
Plane Figure DrawingPerimeter/
Circumference
Triangle
Parallelogram
Rectangle
Square
Trapezoid
Circle C d or 2 r
26Martin-Gay, Prealgebra, 5ed
Area, Volume, and Surface Area, Volume, and Surface AreaArea
Section 9.3
Area of a RectangleArea of a Rectangle
AreaArea is measured in square units. A is measured in square units. A square unit is a square one unit on each square unit is a square one unit on each side.side.
2
3
For example, start with a rectangle with For example, start with a rectangle with length (length (ll) ) 3 units and width ( units and width (ww) ) 2 units. units.
A l • w
A 3 • 2 units2
A 6 units228
3 • 2 6 A bh
The diagonal of
a parallelogram
forms 2
congruent triangles.
3
2
2
3
3 • 2 6
A lw
RectangleRectangle
3
2
TriangleTriangle
Area Area FormulasFormulas
1A bh2
(3 • 2) 3 1
2
ParallelogramParallelogram
3
2
29Martin-Gay, Prealgebra, 5ed
area side • side
A s • s s
2
More Area More Area FormulasFormulas
B
B
bb+
h
h
SquareSquare
side
side
TrapezoidTrapezoid
1( )
2B b
1( )
2A B b
30Martin-Gay, Prealgebra, 5ed
AreaArea is always measured in square units. is always measured in square units.
When finding the area of figures, check to When finding the area of figures, check to make sure that all measurements are the make sure that all measurements are the same units before calculations are made.same units before calculations are made.
Helpful HintHelpful Hint
31Martin-Gay, Prealgebra, 5ed
r
Given a circle of radius, Given a circle of radius, r,, the the circumference iscircumference is C 2 r.
32Martin-Gay, Prealgebra, 5ed
Area of a CircleArea of a Circle
33Martin-Gay, Prealgebra, 5ed
Notice the rectangular shape.
A lw
A (r)r
A r
2
equal sectorsn
2
r
r
r
64 equal sectors
r
r
r
128 equal sectors
r
r
r
Area of a CircleArea of a Circle
34Martin-Gay, Prealgebra, 5ed
P a b c
P a b c d
P 2l 2w
P 4s
P a b c d
Plane Figure DrawingPerimeter/
Circumference Area
Triangle
Parallelogram
Rectangle
Square
Trapezoid
Circle C d or 2 r
A bh
A lw
A s2
A r 2
1( )
2A B b h
1
2A bh
35Martin-Gay, Prealgebra, 5ed
VolumeVolume measures the number of cubic measures the number of cubic units that fill the space of a solid. The units that fill the space of a solid. The volume of a box or can is the amount of volume of a box or can is the amount of space inside.space inside.
Volume can be used to describe the Volume can be used to describe the amount of juice in a pitcher or the amount amount of juice in a pitcher or the amount of concrete needed to pour a foundation for of concrete needed to pour a foundation for a house.a house.
VolumeVolume
36Martin-Gay, Prealgebra, 5ed
A A polyhedron polyhedron is a solid formed by the is a solid formed by the intersection of a finite number of planes.intersection of a finite number of planes.
Surface AreaSurface Area
37
The The surface area surface area of a polyhedron is the of a polyhedron is the sum of the areas of the faces of the sum of the areas of the faces of the polyhedron.polyhedron.
Surface area is measured in square units.Surface area is measured in square units.
Martin-Gay, Prealgebra, 5ed
The volume of a solid is the number of The volume of a solid is the number of cubic unitscubic units in the solid. in the solid.
1 centimeter1 centimeter
1 centimeter
1 inch1 inch
1 inch
1 cubic centimeter 1 cubic inch
38Martin-Gay, Prealgebra, 5ed
Rectangular SolidRectangular Solid
length
widthheight
Volume length width height
V = lwh
SA = 2lh + 2wh + 2lw
39Martin-Gay, Prealgebra, 5ed
CubeCube
sideside
side
Volume = side side side
V = s3
SA = 6s3
40Martin-Gay, Prealgebra, 5ed
SphereSphere
3( adi4
r3
us) Volume
radius
34
3V r
41
24SA r
Martin-Gay, Prealgebra, 5ed
Circular CylinderCircular Cylinder
Volume
V r2h
radius
height
2( adius) ( ht)r heig
42
SA 2 r h + 2r2
Martin-Gay, Prealgebra, 5ed
ConeCone
Volume
radius
height
21
3V r h
2( adius) (1
r hei3
ght)
43 2 2 2S hA r rr
Square-Based PyramidSquare-Based Pyramid
Volume
height
side13
2 ( )side height
V = 13
s2h
44
1
2SA B pl
B = area of base, p = perimeter, l = slant height
Helpful HintHelpful Hint
Volume is always measured in Volume is always measured in cubic unitscubic units..
45
Surface area is always measured in Surface area is always measured in square square unitsunits..
Martin-Gay, Prealgebra, 5ed
Linear MeasurementLinear Measurement
Section 9.4
The U.S. system of measurement uses the The U.S. system of measurement uses the inchinch, , footfoot, , yardyard,, and and milemile to measure to measure lengthlength..
U.S. Units of LengthU.S. Units of Length
12 inches (in.) 1 foot (ft)
3 feet 1 yard (yd)
5280 feet 1 mile (mi)
Unit FractionsUnit Fractions
12 in. 1 ft1
1ft 12 in.
3 ft 1 yd1
1 yd 3 ft
5280 ft 1mi1
1mi 5280 ft
47Martin-Gay, Prealgebra, 5ed
To convert from one unit of length to To convert from one unit of length to another, another, unit fractionsunit fractions may be used. A unit may be used. A unit fraction is a fraction that equals 1.fraction is a fraction that equals 1.
To convert 60 inches to feet, To convert 60 inches to feet, multiply by a unit multiply by a unit fractionfraction that relates feet to inches. The unit fraction that relates feet to inches. The unit fraction should be written so that should be written so that the units we are converting the units we are converting to,to, feet, feet, are in the numerator and the original units,are in the numerator and the original units, inches, inches, are in the denominatorare in the denominator. .
Unit fractionunits converting to
original units
60 in. 1 ft
1 12 in.60 in. 60 1ft
1 12
60 ft5 ft
12
48Martin-Gay, Prealgebra, 5ed
The basic unit of length in the metric The basic unit of length in the metric system is the system is the metermeter. A meter is slightly . A meter is slightly longer than a yard. It is approximately 39.37 longer than a yard. It is approximately 39.37 inches long. Like the decimal system, the inches long. Like the decimal system, the metric system uses powers of ten to define metric system uses powers of ten to define units.units.
49Martin-Gay, Prealgebra, 5ed
Metric System of Measurement
Prefix kilo hecto deka
deci centi milli
Meaning 1000
100 10
1/10 1/100
1/1000 1 millimeter (mm) 1/1000 or 0.001 m
1 centimeter (cm) 1/100 or 0.01 m
1 decimeter (dm) 1/10 or 0.1 m
1 meter (m) 1 m 1 dekameter (dam) 10 m
1 hectometer (hm) 100 m
1 kilometer (km) 1000 meters (m)
Metric Unit of Length
50Martin-Gay, Prealgebra, 5ed
The most commonly used measurements of The most commonly used measurements of length in the metric system are the length in the metric system are the metermeter, , millimetermillimeter,, centimeter centimeter, and , and kilometerkilometer..
51Martin-Gay, Prealgebra, 5ed
As with the U.S. system of measurement, As with the U.S. system of measurement, unit fractions may be used to convert from unit fractions may be used to convert from one unit of length to another. The major one unit of length to another. The major advantage of the metric system is the ease advantage of the metric system is the ease of converting from one unit of length to of converting from one unit of length to another. Since all units of length are powers another. Since all units of length are powers of 10 of the meter, converting from one unit of 10 of the meter, converting from one unit of length to another is as simple as moving of length to another is as simple as moving the decimal point. the decimal point.
52Martin-Gay, Prealgebra, 5ed
Listing units of length in order from largest Listing units of length in order from largest to smallest helps keep track of how many to smallest helps keep track of how many places to move the decimal point when places to move the decimal point when converting.converting.
km hm dam m dm cm mm
Using the listing of units of length, convert Using the listing of units of length, convert 3.5 m to centimeters.3.5 m to centimeters.
Start End
2 units to the right
3.50 m = 350. cm or 350 cm
2 places to the right
53Martin-Gay, Prealgebra, 5ed
Weight and MassWeight and Mass
Section 9.5
Whenever we talk about how heavy an Whenever we talk about how heavy an object is, we are concerned with theobject is, we are concerned with the object’sobject’s weightweight. We discuss weight when . We discuss weight when we refer to a 12-ounce box of cereal, an we refer to a 12-ounce box of cereal, an overweight 19-pound tabby cat, or a overweight 19-pound tabby cat, or a barge hauling 24 tons of garbage.barge hauling 24 tons of garbage.
55Martin-Gay, Prealgebra, 5ed
U.S. Units of WeightU.S. Units of Weight Unit FractionsUnit Fractions
16 ounces (oz) 1 pound (lb)
2000 pounds 1 ton
The U.S. system of measurement uses theThe U.S. system of measurement uses the ounceounce, , poundpound, and , and ton ton to measure weight.to measure weight.
2000 lb 1 ton1
1ton 2000 lb
16 oz 1lb1
1lb 16 oz
56Martin-Gay, Prealgebra, 5ed
In scientific and technical areas, a careful In scientific and technical areas, a careful distinction is made between distinction is made between weightweight and and massmass.. WeightWeight is really a measure of the pull of is really a measure of the pull of gravity. The farther from Earth an object gets, gravity. The farther from Earth an object gets, the less it weighs. the less it weighs. MassMass is a measure of the is a measure of the amount of substance in the object and does amount of substance in the object and does not change. Astronauts orbiting Earth weigh not change. Astronauts orbiting Earth weigh much less than they weigh on Earth, but they much less than they weigh on Earth, but they have the same mass in orbit as they do on have the same mass in orbit as they do on Earth.Earth.
MassMass
57Martin-Gay, Prealgebra, 5ed
The basic unit of mass in the metric system is the The basic unit of mass in the metric system is the gramgram. It is defined as the mass of water contained . It is defined as the mass of water contained in a cube 1 centimeter (cm) on each side.in a cube 1 centimeter (cm) on each side.
1 cm1 cm
1 cm
A tablet contains 200 milligrams of ibuprofen.
A large paper clip weighs approximately 1 gram.
A box of crackers weighs 453 grams.
A kilogram is slightly over 2 pounds.
An adult woman may weigh 60 kilograms.
58Martin-Gay, Prealgebra, 5ed
The prefixes for units of mass in the metric The prefixes for units of mass in the metric system are the same as for units of length.system are the same as for units of length.
Metric System of Measurement
Prefix kilo hecto deka
deci centi milli
Meaning 1000
100 10
1/10 1/100
1/1000 1 milligram (mg) 1/1000 or 0.001 g
1 centigram (cg) 1/100 or 0.01 g
1 decigram (dg) 1/10 or 0.1 g
1 gram (g) 1 g 1 dekagram (dag) 10 g 1 hectogram (hg) 100 g
1 kilogram (kg) 1000 grams (g)
Metric Unit of Length
59Martin-Gay, Prealgebra, 5ed
The three most commonly used units of The three most commonly used units of mass in the metric system are the mass in the metric system are the milligrammilligram, the, the gramgram, and the , and the kilogramkilogram..
As with length, all units of mass are powers of 10 As with length, all units of mass are powers of 10 of the gram, so converting from one unit of mass of the gram, so converting from one unit of mass to another only involves moving the decimal point.to another only involves moving the decimal point.
kg hg dag g dg cg mg
StartEnd
2 units to the left
Using the listing of units of mass, convert 4.75 cg to grams.
04.75 cg = 0.0475 g
2 places to the left60
CapacityCapacity
Section 9.6
Units ofUnits of capacitycapacity are generally used to are generally used to measure liquids. The number of gallons of measure liquids. The number of gallons of gasoline needed to fill a gas tank in a car, gasoline needed to fill a gas tank in a car, the number of cups of water needed in a the number of cups of water needed in a bread recipe, and the number of quarts of bread recipe, and the number of quarts of milk sold each day at a supermarket are all milk sold each day at a supermarket are all examples of using units of capacity.examples of using units of capacity.
62Martin-Gay, Prealgebra, 5ed
U.S. Units of CapacityU.S. Units of Capacity Unit FractionsUnit Fractions
8 fluid ounces (fl oz) 1 cup (c)
2 cups 1 pint (pt)
2 pints 1 quart (qt)
4 quarts 1 gallon (g)
8 fl oz 1 c1
1 c 8 fl oz
2 c 1pt1
1pt 2 c
2 pt 1 qt1
1 qt 2 pt
4 qt 1 gal1
1 gal 4 qt
63Martin-Gay, Prealgebra, 5ed
Capacity: Metric System of Capacity: Metric System of MeasurementMeasurement
TheThe liter liter is the basic unit of capacity in the is the basic unit of capacity in the metric system. A liter is the capacity or metric system. A liter is the capacity or volume of a cube measuring 10 centimeters volume of a cube measuring 10 centimeters on each side.on each side.
10 cm10 cm
10 cm
64
The prefixes for metric units of capacity are The prefixes for metric units of capacity are the same as for metric units of length and the same as for metric units of length and mass.mass.
Metric System of Measurement
Prefix kilo hecto deka
deci centi milli
Meaning 1000
100 10
1/10 1/100
1/1000 1 milliliter (ml) 1/1000 or 0.001 L
1 centiliter (cl) 1/100 or 0.01 L
1 deciliter (dl) 1/10 or 0.1 L
1 liter (L) 1 L 1 dekaliter (dal) 10 L 1 hectoliter (hl) 100 L
1 kiloliter (kl) 1000 liters (L)
Metric Unit of Length
65Martin-Gay, Prealgebra, 5ed
As with length and mass, all units of capacity are As with length and mass, all units of capacity are powers of 10 of the liter, so converting from one unit powers of 10 of the liter, so converting from one unit of capacity to another only involves moving the of capacity to another only involves moving the decimal point.decimal point.
The two most commonly used units of The two most commonly used units of capacity in the metric system are the capacity in the metric system are the millilitermilliliter and the and the literliter..
kl hl dal L dl cl ml
StartEnd
3 units to the left
Using the listing of units of capacity, convert 5350 ml to liters. 5350 ml = 5.350 L
3 places to the left66
Martin-Gay, Prealgebra, 5ed
Temperature and Conversions Temperature and Conversions Between the U.S. and Metric Between the U.S. and Metric SystemsSystems
Section 9.7
LengthLength
2.54 cm 1in.
1.61km 1mi
MetricMetric U.S. SystemU.S. System1m 1.09 yd
1m 3.28 ft
1km 0.62 mi
0.30 m 1ft
68Martin-Gay, Prealgebra, 5ed
CapacityCapacity
1L 1.06 qt
1L 0.26 gal
0.95 L 1qt
MetricMetric U.S. SystemU.S. System
29.57 ml 1fl oz
3.79 L 1gal
69Martin-Gay, Prealgebra, 5ed
Weight (Mass)Weight (Mass)
1kg 2.20 lb
1g 0.04 oz
MetricMetric U.S. SystemU.S. System
0.45 kg 1lb
28.35 g 1oz
70Martin-Gay, Prealgebra, 5ed
Converting Celsius to FahrenheitConverting Celsius to Fahrenheit
9
5
9F C 32
5= + F 1.8C 32= +
(To convert to Fahrenheit temperature, (To convert to Fahrenheit temperature,
multiply the Celsius temperature by multiply the Celsius temperature by
or or 1.8, and then add , and then add 32.).)
oror
71Martin-Gay, Prealgebra, 5ed
Converting Fahrenheit to CelsiusConverting Fahrenheit to Celsius
(To convert to Celsius temperature, (To convert to Celsius temperature,
subtract subtract 32 from the Fahrenheit from the Fahrenheit
temperature, and then multiply by .)temperature, and then multiply by .)
5C (F 32)
9= -
5
9
72Martin-Gay, Prealgebra, 5ed