Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 8 Geometry and Measurement

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.1 Lines and Angles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Plane A plane is a flat surface that extends indefinitely. Space extends in all directions indefinitely. Planes

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Points 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

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. AB A B Line AB or AB Ray AB or AB Line Segment AB or AB AB Lines, Segments, and Rays

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Vertex An angle is made up of two rays that share the same endpoint called the vertex. The angle can be named,,, or. The angle can be named  ABC,  CBA,  B, or  x. A B C x The vertex is the middle point. Angles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. An angle can be measured in degrees. There are 360º (degrees) in a full revolution or full circle. 360º Angles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Classifying Angles Acute Angle Name Angle MeasureExamples Between 0° and 90° Right Angle Exactly 90° Obtuse Angle Between 90° and 180° Straight Angle Exactly 180º

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Two angles that have a sum of 90° are called complementary angles. Two angles that have a sum of 180° are called supplementary angles. Classifying Angles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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.” p q Parallel linesIntersecting lines p  q Parallel Lines

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Two lines are perpendicular if they form right angles when they intersect. The symbol  is used to denote “is perpendicular to.” Perpendicular lines n m n  mn  m Perpendicular Lines

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 b c d  m  a  m  cm  d  m  b Vertical Angles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 a b c d Adjacent Angles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A line that intersects two or more lines at different points is called a transversal. Transversal Lines

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Parallel Lines Cut by a Transversal If two parallel lines are cut by a transversal, then the measures of corresponding angles are equal and the measures of the alternate interior angles are equal. a b c d e f gh

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Corresponding angles have equal measures. a b c d e f gh Corresponding angles  m  a  m  e  m  c  m  g  m  b  m  f  m  d  m  h

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 17 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Alternate interior angles are angles on opposite sides of the transversal between the two parallel lines. a b c d e f gh Alternate Interior Angles  m  c  m  f  m  d  m  e

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.2 Perimeter

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 19 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The perimeter of a polygon is the distance around the polygon, which is the sum of the lengths of its sides. Perimeter

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 20 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. llw w P  2l  2w Perimeter of a Rectangle Perimeter  2 length  2 width width length width length

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 21 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. s sss P  4s Perimeter of a Square Perimeter  4 ∙ side side

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. b a c P  a  b  cP  a  b  c Perimeter of a Triangle a b c P  side a  side b  side c

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 23 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The circumference is the distance around a circle. diameter Circumference ÷ diameter always results in the same ratio, π. This number is named “pi” and is approximately (≈) equal to or Circumference of a Circle

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 24 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Circumference of a Circle r d Circumference = 2 π radius Circumference = π diameter or C = 2π r or C = π d

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.3 Area, Volume, and Surface Area

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 26 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Area Formulas Continued.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 27 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Area Formulas

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 28 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Area is always measured in square units. When finding the area of figures, check to make sure that all measurements are in the same units before calculations are made. Helpful Hint

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 29 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. r Given a circle of radius, r, the circumference is C = 2π r. Radius

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 30 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Dividing the circle into 4 equal sectors, half the circumference, πr, is blue and the other half is red. r πrπr πrπr 32 equal sectors r πrπr πrπr 16 equal sectors πrπr πrπr r 8 equal sectors r πrπr πrπr Area of a Circle

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 31 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Notice the rectangular shape. A  lw A  (πr)r A  πr 2 2 n equal sectors πrπr πrπr r 64 equal sectors πrπr πrπr r 128 equal sectors πrπr r πrπr Area of a Circle

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 32 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. P  a  b  c P  a  b  c  dP  a  b  c  d P  2l  2w P  4s P  a  b  c  dP  a  b  c  d Plane FigureDrawing Perimeter/ Circumference Area Triangle Parallelogram Rectangle Square Trapezoid Circle C  π d or 2π r A  bh A  lw A  s2A  s2 A  πr 2A  πr 2 Plane Figures

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Volume measures the number of cubic units that fill the space of a solid. The volume of a box or can is the amount of space inside. Volume can be used to describe the amount of juice in a pitcher or the amount of concrete needed to pour a foundation for a house. Volume

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 34 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A polyhedron is a solid formed by the intersection of a finite number of planes. Surface Area The surface area of a polyhedron is the sum of the areas of the faces of the polyhedron. Surface area is measured in square units.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 35 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The volume of a solid is the number of cubic units in the solid. 1 centimeter 1 inch 1 cubic centimeter 1 cubic inch Volume

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 36 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Rectangular Solid length width height Volume = length ∙ width ∙ height V = lwh SA = 2lh + 2wh + 2lw

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 37 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Cube side Volume = side ∙ side ∙ side V = s 3 SA = 6s 3

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 38 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Sphere Volume  radius

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 39 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Circular Cylinder Volume  V  πr 2 h radius height SA  2πrh + 2πr 2

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 40 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Cone Volume  radius height

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 41 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Square-Based Pyramid height side B = area of base, p = perimeter, l = slant height

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.4 Linear Measurement

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 43 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The U.S. system of measurement uses the inch, foot, yard, and mile to measure length. U.S. Units of Length 12 inches (in.) = 1 foot (ft) 3 feet = 1 yard (yd) 36 inches = 1 yard 5280 feet = 1 mile (mi) Unit Fractions U.S. System of Measurement

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To convert from one unit of length to another, unit fractions may be used. A unit fraction is a fraction that equals 1. To convert 60 inches to feet, multiply by a unit fraction that relates feet to inches. The unit fraction should be written so that the units we are converting to, feet, are in the numerator and the original units, inches, are in the denominator. Unit fraction units converting to original units 60 in. Converting Between Measurements

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 45 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The basic unit of length in the metric system is the meter. A meter is slightly longer than a yard. It is approximately inches long. Like the decimal system, the metric system uses powers of ten to define units. Metric System of Measurement

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 46 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Metric System of Measurement Prefix kilo hecto deka deci centi milli Meaning /10 1/100 1/ millimeter (mm)  1/1000 or 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 Metric System of Measurement

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 47 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The most commonly used measurements of length in the metric system are the meter, millimeter, centimeter, and kilometer. Metric System of Measurement

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 48 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. As with the U.S. system of measurement, unit fractions may be used to convert from one unit of length to another. The major advantage of the metric system is the ease of converting from one unit of length to another. Since all units of length are powers of 10 of the meter, converting from one unit of length to another is as simple as moving the decimal point. Converting Between Measurements

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 49 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Listing units of length in order from largest to smallest helps keep track of how many places to move the decimal point when converting. km hm dam m dm cm mm Using the listing of units of length, convert 3.5 m to centimeters. StartEnd 2 units to the right 3.50 m= 350. cm or 350 cm 2 places to the right Converting Between Measurements

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.5 Weight and Mass

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 51 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Whenever we talk about how heavy an object is, we are concerned with the object’s weight. We discuss weight when we refer to a 12-ounce box of cereal, an overweight 19-pound tabby cat, or a barge hauling 24 tons of garbage. Weight

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 52 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. U.S. Units of WeightUnit Fractions 16 ounces (oz) = 1 pound (lb) 2000 pounds = 1 ton The U.S. system of measurement uses the ounce, pound, and ton to measure weight. Weight

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 53 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In scientific and technical areas, a careful distinction is made between weight and mass. Weight is really a measure of the pull of gravity. The farther from Earth an object gets, the less it weighs. However, mass is a measure of the amount of substance in the object and does not change. Astronauts orbiting Earth weigh much less than they weigh on Earth, but they have the same mass in orbit as they do on Earth. Mass

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 54 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The basic unit of mass in the metric system is the gram. It is defined as the mass of water contained in a cube 1 centimeter (cm) on each side. 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. Mass - U.S. System

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The prefixes for units of mass in the metric system are the same as for units of length. Metric System of Measurement Prefix kilo hecto deka deci centi milli Meaning /10 1/100 1/ milligram (mg)  1/1000 or 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 Mass – Metric System

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 56 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The three most commonly used units of mass in the metric system are the milligram, the gram, and the kilogram. As with length, all units of mass are powers of 10 of the gram, so converting from one unit of mass 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 cg= g 2 places to the left Mass – Metric System

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.6 Capacity

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 58 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Units of capacity are generally used to measure liquids. The number of gallons of gasoline needed to fill a gas tank in a car, the number of cups of water needed in a bread recipe, and the number of quarts of milk sold each day at a supermarket are all examples of using units of capacity. Capacity

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 59 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. U.S. Units of CapacityUnit Fractions 8 fluid ounces (fl oz)  1 cup (c) 2 cups  1 pint (pt) 2 pints  1 quart (qt) 4 quarts  1 gallon (g) Capacity - U.S. Units

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 60 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The liter is the basic unit of capacity in the metric system. A liter is the capacity or volume of a cube measuring 10 centimeters on each side. 10 cm Capacity - Metric Units

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 61 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The prefixes for metric units of capacity are the same as for metric units of length and mass. Metric System of Measurement Prefix kilo hecto deka deci centi milli Meaning /10 1/100 1/ milliliter (ml)  1/1000 or 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 Capacity - Metric Units

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 62 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. As with length and mass, all units of capacity are powers of 10 of the liter, so converting from one unit of capacity to another only involves moving the decimal point. The two most commonly used units of capacity in the metric system are the milliliter and the liter. kl hl dal L dl cl ml Start End 3 units to the left Using the listing of units of capacity, convert 5350 ml to liters ml= L 3 places to the left Capacity - Metric Units

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 8.7 Temperature and Conversions Between the U.S. and Metric Systems

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 64 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Length MetricU.S. System =

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 65 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Capacity MetricU.S. System

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Weight (Mass) MetricU.S. System

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 67 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Converting Celsius to Fahrenheit (To convert to Fahrenheit temperature, multiply the Celsius temperature by or 1.8, and then add 32.) or

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 68 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Converting Fahrenheit to Celsius (To convert to Celsius temperature, subtract 32 from the Fahrenheit temperature, and then multiply by.)