Geometry and Measurement Chapter Nine

Lines and Angles Section 9.1

Plane A plane is a flat surface that extends indefinitely. Space extends in all directions indefinitely. 3 Martin-Gay, Prealgebra, 5ed

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 4 Martin-Gay, Prealgebra, 5ed

AB A B Line AB or AB Ray AB or AB Line Segment AB or AB AB 5 Martin-Gay, Prealgebra, 5ed

Vertex An angle is made up of two rays that share the same endpoint called a 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. 6

An angle can be measured in degrees. There are 360º (degrees) in a full revolution or full circle. 360º 7 Martin-Gay, Prealgebra, 5ed

Classifying Angles Acute Angle Name Angle Measure Examples Between 0° and 90° Right Angle Exactly 90° Obtuse Angle Between 90° and 180° Straight Angle Exactly 180º 8

Two angles that have a sum of 90° are called complementary angles. Two angles that have a sum of 180° are called supplementary angles. 9 Martin-Gay, Prealgebra, 5ed

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 10 Martin-Gay, Prealgebra, 5ed

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 11 Martin-Gay, Prealgebra, 5ed

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 a  cd  b 12 Martin-Gay, Prealgebra, 5ed

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 13 Martin-Gay, Prealgebra, 5ed

A line that intersects two or more lines at different points is called a transversal. 14 Martin-Gay, Prealgebra, 5ed

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. 15 Martin-Gay, Prealgebra, 5ed

Corresponding angles are equal. a b c d e f gh 16 Martin-Gay, Prealgebra, 5ed

Alternate interior angles are angles on opposite sides of the transversal between the two parallel lines. a b c d e f gh 17 Martin-Gay, Prealgebra, 5ed

Perimeter Section 9.2

Two measures of plane figures are important: the distance around a plane figure called the perimeter or circumferencethe distance around a plane figure called the perimeter or circumference the number of square units in the interior of a plane figure called the area.the number of square units in the interior of a plane figure called the area. and 19 Martin-Gay, Prealgebra, 5ed

llw w P  2l  2w Perimeter of a Rectangle Perimeter  2 length  2 width width length width length 20 Martin-Gay, Prealgebra, 5ed

Perimeter is always measured in units. Helpful Hint 21 Martin-Gay, Prealgebra, 5ed

s sss P  4s Perimeter of a Square Perimeter  4 side side 22 Martin-Gay, Prealgebra, 5ed

b a c P  a  b  cP  a  b  c Perimeter of a Triangle The perimeter of every polygon may be found by adding all the sides. a b c P  side a  side b  side c 23

The circumference is the distance around a circle. Circumference  diameter always results in the same ratio. This number is named “pi” and is approximately (  ) equal to or diameter 24 Martin-Gay, Prealgebra, 5ed

Circumference of a Circle r d Circumference  2·   ·radius Circumference    ·diameter or C  2   r or C    d or 25 Martin-Gay, Prealgebra, 5ed

26 Martin-Gay, Prealgebra, 5ed

Area, Volume, and Surface Area Section 9.3

Area of a Rectangle Area is measured in square units. A square unit is a square one unit on each side. 2 3 For example, start with a rectangle with length (l) units and width (w) units. For example, start with a rectangle with length (l) 3 units and width (w) 2 units. A  l w A  3 2 units 2 A  6 units 2 28

3 2  6 A  bh The diagonal of a parallelogram forms 2 congruent triangles  6 A  lwRectangle 3 2 Triangle Area Formulas 1 A  bh 2 (3 2)  3 Parallelogram Martin-Gay, Prealgebra, 5ed

area  side side A  s s  s 2A  s s  s 2 More Area Formulas B B b b + h h Square side Trapezoid 30 Martin-Gay, Prealgebra, 5ed

Area is always measured in square units. When finding the area of figures, check to make sure that all measurements are the same units before calculations are made. Helpful Hint 31 Martin-Gay, Prealgebra, 5ed

r Given a circle of radius,, the circumference is Given a circle of radius, r, the circumference is C  2  r. 32 Martin-Gay, Prealgebra, 5ed

Area of a Circle 33 Martin-Gay, Prealgebra, 5ed

Notice the rectangular shape. A  lw A  (  r)r A   r 2 equal sectors n 2 rr rr r 64 equal sectors rr rr r 128 equal sectors rr r rr Area of a Circle 34 Martin-Gay, Prealgebra, 5ed

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   d or 2  r A  bh A  lw A  s2A  s2 A  r 2A  r 2 35 Martin-Gay, Prealgebra, 5ed

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 36 Martin-Gay, Prealgebra, 5ed

A polyhedron is a solid formed by the intersection of a finite number of planes. Surface Area 37 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, 5ed

The volume of a solid is the number of cubic units in the solid. 1 centimeter 1 inch 1 cubic centimeter1 cubic inch 38 Martin-Gay, Prealgebra, 5ed

Rectangular Solid length width height Volume  length  width  height V = lwh SA = 2lh + 2wh + 2lw 39 Martin-Gay, Prealgebra, 5ed

Cube side Volume = side  side  side V = s 3 SA = 6s 3 40 Martin-Gay, Prealgebra, 5ed

Sphere Volume  radius 41 Martin-Gay, Prealgebra, 5ed

Circular Cylinder Volume  V   r 2 h radius height 42 SA  2  r h + 2  r 2 Martin-Gay, Prealgebra, 5ed

Cone Volume  radius height 43

Square-Based Pyramid Volume  height side  ()‏ sideheight V = 1 3 s2hs2h 44 B = area of base, p = perimeter, l = slant height

Helpful Hint Volume is always measured in cubic units. 45 Surface area is always measured in square units. Martin-Gay, Prealgebra, 5ed

Linear Measurement Section 9.4

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)‏ 5280 feet  1 mile (mi)‏ Unit Fractions 47 Martin-Gay, Prealgebra, 5ed

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. 48 Martin-Gay, Prealgebra, 5ed

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. 49 Martin-Gay, Prealgebra, 5ed

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 50 Martin-Gay, Prealgebra, 5ed

The most commonly used measurements of length in the metric system are the meter, millimeter, centimeter, and kilometer. 51 Martin-Gay, Prealgebra, 5ed

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. 52 Martin-Gay, Prealgebra, 5ed

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 53 Martin-Gay, Prealgebra, 5ed

Weight and Mass Section 9.5

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. 55 Martin-Gay, Prealgebra, 5ed

U.S. Units of Weight Unit 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. 56 Martin-Gay, Prealgebra, 5ed

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. 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 57 Martin-Gay, Prealgebra, 5ed

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. 58 Martin-Gay, Prealgebra, 5ed

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 59 Martin-Gay, Prealgebra, 5ed

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 60

Capacity Section 9.6

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. 62 Martin-Gay, Prealgebra, 5ed

U.S. Units of Capacity Unit Fractions 8 fluid ounces (fl oz)  1 cup (c)‏ 2 cups  1 pint (pt)‏ 2 pints  1 quart (qt)‏ 4 quarts  1 gallon (g)‏ 63 Martin-Gay, Prealgebra, 5ed

Capacity: Metric System of Measurement 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 64

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 65 Martin-Gay, Prealgebra, 5ed

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 StartEnd 3 units to the left Using the listing of units of capacity, convert 5350 ml to liters ml= L 3 places to the left 66 Martin-Gay, Prealgebra, 5ed

Temperature and Conversions Between the U.S. and Metric Systems Section 9.7

Length Metric U.S. System 68 Martin-Gay, Prealgebra, 5ed

Capacity Metric U.S. System 69 Martin-Gay, Prealgebra, 5ed

Weight (Mass)‏ Metric U.S. System 70 Martin-Gay, Prealgebra, 5ed

Converting Celsius to Fahrenheit (To convert to Fahrenheit temperature, multiply the Celsius temperature by or, and then add.)‏ (To convert to Fahrenheit temperature, multiply the Celsius temperature by or 1.8, and then add 32.)‏ or 71 Martin-Gay, Prealgebra, 5ed

Converting Fahrenheit to Celsius (To convert to Celsius temperature, subtract from the Fahrenheit temperature, and then multiply by.)‏ (To convert to Celsius temperature, subtract 32 from the Fahrenheit temperature, and then multiply by.)‏ 72 Martin-Gay, Prealgebra, 5ed