Geometry and Measurement Chapter Nine. Lines and Angles Section 9.1.

Geometry and MeasurementGeometry and Measurement

Chapter NineChapter Nine

Lines and AnglesLines and Angles

Section 9.1

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.

Point A

Line AB or AB

Ray AB or AB

Line Segment AB or AB

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. .

The vertex is themiddle point.

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.

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..

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.”

Parallel lines Intersecting linesp q

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

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 c d b

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 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..

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..

Corresponding angles are equal.Corresponding angles are equal.

a bc d

e fg h

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.

a bc d

e fg h

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

ll w w

P 2l 2w

Perimeter of a RectanglePerimeter of a Rectangle

Perimeter 2 • length 2 • width

length

Perimeter is always measured in units.Perimeter is always measured in units.

Helpful HintHelpful Hint

ss s s

Perimeter of a SquarePerimeter of a Square

Perimeter 4 • side

sideside

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.

P side a side b side c

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

diameter

Circumference of a CircleCircumference of a Circle

Circumference 2··radius

Circumference ·diameter

C 2r oror C d

P a b c

P a b c d

P 2l 2w

P a b c d

Plane Figure DrawingPerimeter/

Circumference

Triangle

Parallelogram

Rectangle

Square

Trapezoid

Circle C d or 2 r

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.

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 6

RectangleRectangle

TriangleTriangle

Area Area FormulasFormulas

1A bh2

(3 • 2) 3 1

ParallelogramParallelogram

area side • side

A s • s s

More Area More Area FormulasFormulas

SquareSquare

TrapezoidTrapezoid

2A B b

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.

Given a circle of radius, Given a circle of radius, r,, the the circumference iscircumference is C 2 r.

Area of a CircleArea of a Circle

Notice the rectangular shape.

A (r)r

equal sectorsn

64 equal sectors

128 equal sectors

Area of a CircleArea of a Circle

P a b c

P a b c d

P 2l 2w

P a b c d

Plane Figure DrawingPerimeter/

Circumference Area

Triangle

Parallelogram

Rectangle

Square

Trapezoid

Circle C d or 2 r

2A B b h

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

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

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.

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

Rectangular SolidRectangular Solid

length

widthheight

Volume length width height

V = lwh

SA = 2lh + 2wh + 2lw

CubeCube

sideside

Volume = side side side

V = s3

SA = 6s3

SphereSphere

3( adi4

us) Volume

radius

24SA r

Circular CylinderCircular Cylinder

Volume

radius

height

2( adius) ( ht)r heig

SA 2 r h + 2r2

ConeCone

Volume

radius

height

3V r h

2( adius) (1

r hei3

43 2 2 2S hA r rr

Square-Based PyramidSquare-Based Pyramid

Volume

height

side13

2 ( )side height

V = 13

2SA B pl

B = area of base, p = perimeter, l = slant height

Volume is always measured in Volume is always measured in cubic unitscubic units..

Surface area is always measured in Surface area is always measured in square square unitsunits..

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

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

60 ft5 ft

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.

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

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..

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.

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

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.

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

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

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

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.

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.

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)

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.

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

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

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.

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)

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

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

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

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

Converting Celsius to FahrenheitConverting Celsius to Fahrenheit

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.).)

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)

Geometry and Measurement Chapter Nine. Lines and Angles Section 9.1.

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