CHAPTER 8 Geometry Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 8.1Basic Geometric Figures 8.2Perimeter 8.3Area 8.4Circles 8.5Volume and Surface Area 8.6Relationships Between Angle Measures 8.7Congruent Triangles and Properties of Parallelograms 8.8Similar Triangles

OBJECTIVES 8.1 Basic Geometric Figures Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aDraw and name segments, rays, and lines. Also, identify endpoints, if they exist. bName an angle in five different ways, and given an angle, measure it with a protractor. cClassify an angle as right, straight, acute, or obtuse. dIdentify perpendicular lines.

OBJECTIVES 8.1 Basic Geometric Figures Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. eClassify a triangle as equilateral, isosceles, or scalene and as right, obtuse, or acute. Given a polygon of twelve, ten, or fewer sides, classify it as a dodecagon, a decagon, and so on. fGiven a polygon of n sides, find the sum of its angle measures using the formula (n – 2) 80.

8.1 Basic Geometric Figures a Draw and name segments, rays, and lines. Also, identify endpoints, if they exist. Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A segment is a geometric figure consisting of two points, called endpoints, and all points between them. The segment whose endpoints are A and B is shown below. It can be named

A ray consists of a segment, say and all points X such that B is between A and X, that is, and all points “beyond” B. 8.1 Basic Geometric Figures a Draw and name segments, rays, and lines. Also, identify endpoints, if they exist. Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures a Draw and name segments, rays, and lines. Also, identify endpoints, if they exist. Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures a Draw and name segments, rays, and lines. Also, identify endpoints, if they exist. Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Lines in the same plane are called coplanar. Coplanar lines that do not intersect are called parallel. For example, lines l and m below are parallel

8.1 Basic Geometric Figures a Draw and name segments, rays, and lines. Also, identify endpoints, if they exist. Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The figure below shows two lines that cross. Their intersection is D. They are also called intersecting lines.

8.1 Basic Geometric Figures b Name an angle in five different ways, and given an angle, measure it with a protractor. Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. An angle is a set of points consisting of two rays, or half- lines, with a common endpoint. The endpoint is called the vertex of the angle. The rays are called the sides of the angle.

8.1 Basic Geometric Figures b Name an angle in five different ways, and given an angle, measure it with a protractor. Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures b Name an angle in five different ways, and given an angle, measure it with a protractor. Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Measuring angles is similar to measuring segments. The unit most commonly used for angle measure is the degree. Below is such a unit. Its measure is 1 degree, or 1°.

8.1 Basic Geometric Figures b Name an angle in five different ways, and given an angle, measure it with a protractor. Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures b Name an angle in five different ways, and given an angle, measure it with a protractor. Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A device called a protractor is used to measure angles. Protractors have two scales. To measure an angle like below, we place the protractor’s at the vertex and line up one of the angle’s sides at Then we check where the angle’s other side crosses the scale. In the figure, is on the inside scale, so we check where the angle’s other side crosses the inside scale. We see that m ∠ Q = 145°. The notation m ∠ Q is read “the measure of angle Q.”

8.1 Basic Geometric Figures b Name an angle in five different ways, and given an angle, measure it with a protractor. Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures b Name an angle in five different ways, and given an angle, measure it with a protractor. Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures TYPES OF ANGLES Slide 17Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Right angle: An angle whose measure is 90. Straight angle: An angle whose measure is 180. Acute angle: An angle whose measure is greater than and less than 90. Obtuse angle: An angle whose measure is greater than and less than 180°.

8.1 Basic Geometric Figures c Classify an angle as right, straight, acute, or obtuse. Slide 18Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures c Classify an angle as right, straight, acute, or obtuse. Slide 19Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures d Identify perpendicular lines. Slide 20Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures e Classify a triangle as equilateral, isosceles, or scalene and as right, obtuse, or acute. Given a polygon of twelve, ten, or fewer sides, classify it as a dodecagon, a decagon, and so on. Slide 21Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The figures below are examples of polygons.

8.1 Basic Geometric Figures e Classify a triangle as equilateral, isosceles, or scalene and as right, obtuse, or acute. Given a polygon of twelve, ten, or fewer sides, classify it as a dodecagon, a decagon, and so on. Slide 22Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A triangle is a polygon made up of three segments, or sides.

8.1 Basic Geometric Figures TYPES OF TRIANGLES Slide 23Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Equilateral triangle: All sides are the same length. Isosceles triangle: Two or more sides are the same length. Scalene triangle: All sides are of different lengths. Right triangle: One angle is a right angle. Obtuse triangle: One angle is an obtuse angle. Acute triangle: All three angles are acute.

8.1 Basic Geometric Figures e Classify a triangle as equilateral, isosceles, or scalene and as right, obtuse, or acute. Given a polygon of twelve, ten, or fewer sides, classify it as a dodecagon, a decagon, and so on. Slide 24Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures SUM OF THE ANGLE MEASURES OF A TRIANGLE Slide 25Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.1 Basic Geometric Figures f Given a polygon of n sides, find the sum of its angle measures using the formula (n – 2) Slide 26Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the missing angle measure.

EXAMPLE 8.1 Basic Geometric Figures f Given a polygon of n sides, find the sum of its angle measures using the formula (n – 2) Slide 27Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8.1 Basic Geometric Figures SUM OF ANGLE MEASURES Slide 28Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.1 Basic Geometric Figures f Given a polygon of n sides, find the sum of its angle measures using the formula (n – 2) Slide 29Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. What is the sum of the angle measures of a hexagon?

EXAMPLE 8.2 Perimeter Slide 30Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aFind the perimeter of a polygon. bSolve applied problems involving perimeter.

EXAMPLE 8.2 Perimeter PERIMETER OF A POLYGON Slide 31Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A polygon is a closed geometric figure with three or more sides. The perimeter of a polygon is the distance around it, or the sum of the lengths of its sides.

EXAMPLE 8.2 Perimeter PERIMETER OF A RECTANGLE Slide 32Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The perimeter of a rectangle is twice the sum of the length and the width, or 2 times the length plus 2 times the width:

EXAMPLE 8.2 Perimeter PERIMETER OF A RECTANGLE Slide 33Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The perimeter of a rectangle is twice the sum of the length and the width, or 2 times the length plus 2 times the width:

EXAMPLE 8.2 Perimeter a Find the perimeter of a polygon. 3 Slide 34Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the perimeter of a rectangle that is 7.8 ft by 4.3 ft.

EXAMPLE 8.2 Perimeter PERIMETER OF A SQUARE Slide 35Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The perimeter of a square is four times the length of a side:

EXAMPLE 8.2 Perimeter b Solve applied problems involving perimeter. 6 Slide 36Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Jaci is adding crown molding to the top edge of each wall in her rectangular dining room, which measures 14 ft by 12 ft. How many feet of trim will be needed? If the crown molding sells for $2.25 per foot, what will the trim cost?

EXAMPLE 8.2 Perimeter b Solve applied problems involving perimeter. 6 Slide 37Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 1. Familiarize. We make a drawing and let P = the perimeter.

EXAMPLE 8.2 Perimeter b Solve applied problems involving perimeter. 6 Slide 38Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 3. Solve. 2. Translate.

EXAMPLE 8.2 Perimeter b Solve applied problems involving perimeter. 6 Slide 39Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 4. Check. The check is left to the student. 5. State. The 52 ft of crown molding that is needed will cost $117.

EXAMPLE 8.3 Area Slide 40Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aFind the area of a rectangle and a square. bFind the area of a parallelogram, a triangle, and a trapezoid. cSolve applied problems involving areas of rectangles, squares, parallelograms, triangles, and trapezoids.

EXAMPLE 8.3 Area AREA OF A RECTANGLE Slide 41Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The area of a rectangle is the product of the length l and the width w:

EXAMPLE 8.3 Area a Find the area of a rectangle and a square. 2 Slide 42Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the area of a rectangle that is 7 yd by 4 yd.

EXAMPLE 8.3 Area AREA OF A SQUARE Slide 43Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The area of a square is the square of the length of a side:

EXAMPLE 8.3 Area a Find the area of a rectangle and a square. 3 Slide 44Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the area of a square with sides of length 9 mm.

EXAMPLE 8.3 Area b Find the area of a parallelogram, a triangle, and a trapezoid. Slide 45Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A parallelogram is a four-sided figure with two pairs of parallel sides.

EXAMPLE 8.3 Area AREA OF A PARALLELOGRAM Slide 46Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The area of a parallelogram is the product of the length of the base b and the height h:

EXAMPLE 8.3 Area b Find the area of a parallelogram, a triangle, and a trapezoid. 5 Slide 47Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the area of this parallelogram.

EXAMPLE 8.3 Area AREA OF A TRIANGLE Slide 48Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The area of a triangle is half the length of the base times the height:

EXAMPLE 8.3 Area b Find the area of a parallelogram, a triangle, and a trapezoid. 8 Slide 49Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the area of this triangle.

EXAMPLE 8.3 Area b Find the area of a parallelogram, a triangle, and a trapezoid. Slide 50Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A trapezoid is a polygon with four sides, two of which, the bases, are parallel to each other.

EXAMPLE 8.3 Area AREA OF A TRAPEZOID Slide 51Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The area of a trapezoid is half the product of the height and the sum of the lengths of the parallel sides (bases):

EXAMPLE 8.3 Area b Find the area of a parallelogram, a triangle, and a trapezoid. 9 Slide 52Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the area of this trapezoid.

EXAMPLE 8.3 Area c Solve applied problems involving areas of rectangles, squares, parallelograms, triangles, and trapezoids. 11Lucas Oil Stadium Slide 53Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The retractable roof of Lucas Oil Stadium, the home of the Indianapolis Colts football team, divides lengthwise. Each half measures 600 ft by 160 ft. The roof opens and closes in approximately 9–11 min. The opening measures 293 ft across. What is the total area of the rectangular roof? What is the area of the opening? Source: HKS Sports and Entertainment

EXAMPLE 8.3 Area c Solve applied problems involving areas of rectangles, squares, parallelograms, triangles, and trapezoids. 11 Slide 54Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.3 Area c Solve applied problems involving areas of rectangles, squares, parallelograms, triangles, and trapezoids. 11 Slide 55Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Each half of the retractable roof is a rectangle that measures 600 ft by 160 ft. The area of a rectangle is length times width, so we have

EXAMPLE 8.3 Area c Solve applied problems involving areas of rectangles, squares, parallelograms, triangles, and trapezoids. 11 Slide 56Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The total area of the two halves of the retractable roof is

EXAMPLE 8.3 Area c Solve applied problems involving areas of rectangles, squares, parallelograms, triangles, and trapezoids. 11 Slide 57Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. When the retractable roof is open, the dimensions of the opening are 600 ft by 293 ft. The area of this rectangle is When the roof is open, the opening is 175,800 ft 2.

EXAMPLE 8.4 Circles Slide 58Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aFind the length of a radius of a circle given the length of a diameter, and find the length of a diameter given the length of a radius. bFind the circumference of a circle given the length of a diameter or a radius. cFind the area of a circle given the length of a diameter or a radius. dSolve applied problems involving circles.

EXAMPLE 8.4 Circles a Find the length of a radius of a circle given the length of a diameter, and find the length of a diameter given the length of a radius. Slide 59Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A diameter is a segment that passes through the center of the circle and has endpoints on the circle. A radius is a segment with one endpoint on the center and the other endpoint on the circle.

EXAMPLE 8.4 Circles DIAMETER AND RADIUS Slide 60Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Suppose that d is the length of the diameter of a circle and r is the length of the radius. Then

EXAMPLE 8.4 Circles a Find the length of a radius of a circle given the length of a diameter, and find the length of a diameter given the length of a radius. 1 Slide 61Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the length of a radius of this circle. The radius is 6 m.

EXAMPLE 8.4 Circles b Find the circumference of a circle given the length of a diameter or a radius. Slide 62Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The circumference of a circle is the distance around it. Calculating circumference is similar to finding the perimeter of a polygon.

EXAMPLE 8.4 Circles CIRCUMFERENCE AND DIAMETER Slide 63Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.4 Circles b Find the circumference of a circle given the length of a diameter or a radius. 3 Slide 64Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.4 Circles CIRCUMFERENCE AND RADIUS Slide 65Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The circumference C of a circle of radius r is given by

EXAMPLE 8.4 Circles b Find the circumference of a circle given the length of a diameter or a radius. 4 Slide 66Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.4 Circles AREA OF A CIRCLE Slide 67Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.4 Circles c Find the area of a circle given the length of a diameter or a radius. 7 Slide 68Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.4 Circles d Solve applied problems involving circles. 8Area of Pizza Pans Slide 69Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. How much larger is a pizza made in a 16-in.-square pizza pan than a pizza made in a 16-in.-diameter circular pan?

EXAMPLE 8.4 Circles d Solve applied problems involving circles. 8Area of Pizza Pans Slide 70Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. First, we make a drawing of each.

EXAMPLE 8.4 Circles d Solve applied problems involving circles. 8Area of Pizza Pans Slide 71Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.4 Circles d Solve applied problems involving circles. 8Area of Pizza Pans Slide 72Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.5 Volume and Surface Area Slide 73Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aFind the volume and the surface area of a rectangular solid. bGiven the radius and the height, find the volume of a circular cylinder. cGiven the radius, find the volume of a sphere. dGiven the radius and the height, find the volume of a circular cone.

EXAMPLE 8.5 Volume and Surface Area Slide 74Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. eSolve applied problems involving volumes of rectangular solids, circular cylinders, spheres, and cones.

EXAMPLE 8.5 Volume and Surface Area a Find the volume and the surface area of a rectangular solid. Slide 75Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The volume of a rectangular solid is the number of unit cubes needed to fill it.

EXAMPLE 8.5 Volume and Surface Area VOLUME OF A RECTANGULAR SOLID Slide 76Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The volume of a rectangular solid is found by multiplying length by width by height:

EXAMPLE 8.5 Volume and Surface Area a Find the volume and the surface area of a rectangular solid. 2Volume of a Safe Slide 77Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. For office security, William purchases a safe whose dimensions are 29 in. by 25 in. by 53 in. Find the volume of this rectangular solid.

EXAMPLE 8.5 Volume and Surface Area a Find the volume and the surface area of a rectangular solid. Slide 78Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The surface area of a rectangular solid is the total area of the six rectangles that form the surface of the solid.

EXAMPLE 8.5 Volume and Surface Area SURFACE AREA OF A RECTANGULAR SOLID Slide 79Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The surface area of a rectangular solid with length l, width w, and height h is given by the formula

EXAMPLE 8.5 Volume and Surface Area a Find the volume and the surface area of a rectangular solid. 3 Slide 80Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the surface area of this rectangular solid.

EXAMPLE 8.5 Volume and Surface Area Slide 81Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The units used for area are square units. The units used for volume are cubic units.

EXAMPLE 8.5 Volume and Surface Area VOLUME OF A CIRCULAR CYLINDER Slide 82Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The volume of a circular cylinder is the product of the area of the base B and the height h:

EXAMPLE 8.5 Volume and Surface Area b Given the radius and the height, find the volume of a circular cylinder. 4 Slide 83Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.5 Volume and Surface Area c Given the radius, find the volume of a sphere. Slide 84Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A sphere is the three-dimensional counterpart of a circle. It is the set of all points in space that are a given distance (the radius) from a given point (the center).

EXAMPLE 8.5 Volume and Surface Area VOLUME OF A SPHERE Slide 85Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.5 Volume and Surface Area c Given the radius, find the volume of a sphere. 6Bowling Ball Slide 86Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The radius of a standard-sized bowling ball is in. Find the volume of a standard-sized bowling ball. Round to the nearest hundredth of a cubic inch. Use 3.14 for

EXAMPLE 8.5 Volume and Surface Area d Given the radius and the height, find the volume of a circular cone. Slide 87Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Consider a circle in a plane and choose any point P not in the plane. The circular region, together with the set of all segments connecting P to a point on the circle, is called a circular cone.

EXAMPLE 8.5 Volume and Surface Area VOLUME OF A CIRCULAR CONE Slide 88Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The volume of a circular cone with base radius r is one- third the product of the area of the base and the height:

EXAMPLE 8.5 Volume and Surface Area d Given the radius and the height, find the volume of a circular cone. 7 Slide 89Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.5 Volume and Surface Area e Solve applied problems involving volumes of rectangular solids, circular cylinders, spheres, and cones. 8Propane Gas Tank Slide 90Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A propane gas tank is shaped like a circular cylinder with half of a sphere at each end. Find the volume of the tank if the cylindrical section is 5 ft long with a 4-ft diameter. Use 3.14 for

EXAMPLE 8.5 Volume and Surface Area e Solve applied problems involving volumes of rectangular solids, circular cylinders, spheres, and cones. 8Propane Gas Tank Slide 91Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 1. Familiarize. We first make a drawing.

EXAMPLE 8.5 Volume and Surface Area e Solve applied problems involving volumes of rectangular solids, circular cylinders, spheres, and cones. 8Propane Gas Tank Slide 92Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 2. Translate. This is a two-step problem. We first find the volume of the cylindrical portion. Then we find the volume of the two ends and add. Note that together the two ends make a sphere with a radius of 2 ft. We let

EXAMPLE 8.5 Volume and Surface Area e Solve applied problems involving volumes of rectangular solids, circular cylinders, spheres, and cones. 8Propane Gas Tank Slide 93Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.5 Volume and Surface Area e Solve applied problems involving volumes of rectangular solids, circular cylinders, spheres, and cones. 8Propane Gas Tank Slide 94Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 3. Solve. The volume of the cylinder is approximately The volume of the two ends is approximately The total volume is about

EXAMPLE 8.5 Volume and Surface Area e Solve applied problems involving volumes of rectangular solids, circular cylinders, spheres, and cones. 8Propane Gas Tank Slide 95Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 4. Check. We can repeat the calculations. The answer checks. 5. State. The volume of the tank is about 96.3 ft 3.

EXAMPLE 8.6 Relationships Between Angle Measures Slide 96Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aIdentify complementary and supplementary angles and find the measure of a complement or a supplement of a given angle. bDetermine whether segments are congruent and whether angles are congruent. cUse the Vertical Angle Property to find measures of angles.

EXAMPLE 8.6 Relationships Between Angle Measures Slide 97Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. dIdentify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles.

EXAMPLE 8.6 Relationships Between Angle Measures COMPLEMENTARY ANGLES Slide 98Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures a Identify complementary and supplementary angles and find the measure of a complement or a supplement of a given angle. Slide 99Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. If two angles are complementary, each is an acute angle. When complementary angles are adjacent to each other, they form a right angle.

EXAMPLE 8.6 Relationships Between Angle Measures a Identify complementary and supplementary angles and find the measure of a complement or a supplement of a given angle. 2 Slide 100Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures SUPPLEMENTARY ANGLES Slide 101Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures a Identify complementary and supplementary angles and find the measure of a complement or a supplement of a given angle. 4 Slide 102Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures b Determine whether segments are congruent and whether angles are congruent. Slide 103Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Congruent figures have the same size and shape. They fit together exactly.

EXAMPLE 8.6 Relationships Between Angle Measures CONGRUENT SEGMENTS Slide 104Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two segments are congruent if and only if they have the same length.

EXAMPLE 8.6 Relationships Between Angle Measures b Determine whether segments are congruent and whether angles are congruent. 6 Slide 105Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Which pairs of segments are congruent? Use a ruler.

EXAMPLE 8.6 Relationships Between Angle Measures CONGRUENT ANGLES Slide 106Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two angles are congruent if and only if they have the same measure.

EXAMPLE 8.6 Relationships Between Angle Measures b Determine whether segments are congruent and whether angles are congruent. 8 Slide 107Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Which pairs of angles are congruent? Use a protractor.

EXAMPLE 8.6 Relationships Between Angle Measures b Determine whether segments are congruent and whether angles are congruent. Slide 108Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. If two angles are congruent, then their supplements are congruent and their complements are congruent.

EXAMPLE 8.6 Relationships Between Angle Measures VERTICAL ANGLES Slide 109Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two nonstraight angles are vertical angles if and only if their sides form two pairs of opposite rays.

EXAMPLE 8.6 Relationships Between Angle Measures THE VERTICAL ANGLE PROPERTY Slide 110Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Vertical angles are congruent.

EXAMPLE 8.6 Relationships Between Angle Measures c Use the Vertical Angle Property to find measures of angles. 9 Slide 111Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures c Use the Vertical Angle Property to find measures of angles. 9 Slide 112Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures TRANSVERSAL Slide 113Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A transversal is a line that intersects two or more coplanar lines in different points.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Slide 114Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Corresponding Angles

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Slide 115Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Interior Angles

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Slide 116Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Alternate Interior Angles

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. 10 Slide 117Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Identify all pairs of corresponding angles, all interior angles, and all pairs of alternate interior angles.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. 10 Slide 118Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Slide 119Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Properties of Parallel Lines 1. If a transversal intersects two parallel lines, then the corresponding angles are congruent.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Slide 120Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 2. If a transversal intersects two parallel lines, then the alternate interior angles are congruent.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Slide 121Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 3. In a plane, if two lines are parallel to a third line, then the two lines are parallel to each other.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Slide 122Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 4. If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. Slide 123Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 5. If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. 11 Slide 124Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. 11 Slide 125Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.6 Relationships Between Angle Measures d Identify pairs of corresponding angles, interior angles, and alternate interior angles and apply properties of transversals and parallel lines to find measures of angles. 12 Slide 126Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms Slide 127Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aIdentify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. bUse properties of parallelograms to find lengths of sides and measures of angles of parallelograms.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. Slide 128Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. Slide 129Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms CONGRUENT TRIANGLES Slide 130Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two triangles are congruent if and only if their vertices can be matched so that the corresponding angles and sides are congruent.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. Slide 131Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The corresponding sides and angles of two congruent triangles are called corresponding parts of congruent triangles. Corresponding parts of congruent triangles are always congruent.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. Slide 132Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. 3 Slide 133Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Suppose that What are the congruent corresponding parts?

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms THE SIDE–ANGLE–SIDE (SAS) PROPERTY Slide 134Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. 5 Slide 135Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Which pairs of triangles on the next slide are congruent by the SAS property? Pairs (b) and (c) are congruent by the SAS property.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. 5 Slide 136Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms THE SIDE–SIDE–SIDE (SSS) PROPERTY Slide 137Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. 6 Slide 138Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Pairs (b) and (d) are congruent by the SSS property. Which pairs of triangles are congruent by the SSS property?

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms THE ANGLE–SIDE–ANGLE (ASA) PROPERTY Slide 139Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. 7 Slide 140Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Which pairs of triangles are congruent by the ASA property? Pairs (b) and (c) are congruent by the ASA property.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. 13 Slide 141Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms a Identify the corresponding parts of congruent triangles and show why triangles are congruent using SAS, SSS, and ASA. 14 Slide 142Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms PROPERTIES OF PARALLELOGRAMS Slide 143Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 1. A diagonal of a parallelogram determines two congruent triangles. 2. The opposite angles of a parallelogram are congruent. 3. The opposite sides of a parallelogram are congruent. 4. Consecutive angles of a parallelogram are supplementary. 5. The diagonals of a parallelogram bisect each other.

EXAMPLE 8.7 Congruent Triangles and Properties of Parallelograms b Use properties of parallelograms to find lengths of sides and measures of angles of parallelograms. 16 Slide 144Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.8 Similar Triangles Slide 145Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aIdentify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. bFind lengths of sides of similar triangles using proportions.

EXAMPLE 8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. Slide 146Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. We know that congruent figures have the same shape and size. Similar figures have the same shape, but are not necessarily the same size.

EXAMPLE 8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. 1 Slide 147Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Which pairs of triangles appear to be similar? Pairs (a), (c), and (d) appear to be similar.

EXAMPLE 8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. Slide 148Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Similar triangles have corresponding sides and angles.

EXAMPLE 8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. 2 Slide 149Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.8 Similar Triangles SIMILAR TRIANGLES Slide 150Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two triangles are similar if and only if their vertices can be matched so that the corresponding angles are congruent and the lengths of corresponding sides are proportional.

EXAMPLE 8.8 Similar Triangles a Identify the corresponding parts of similar triangles and determine which sides of a given pair of triangles have lengths that are proportional. 4 Slide 151Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. These triangles are similar. Which sides are proportional? It appears that if we match X with U, Y with W, and Z with V, the corresponding angles will be congruent. Thus,

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 6 Slide 152Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 6 Slide 153Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Recall that if a transversal intersects two parallel lines, then the alternate interior angles are congruent (Section 6.6). Thus, because they are pairs of alternate interior angles. Since are vertical angles, they are congruent. Thus by definition

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 6 Slide 154Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. and the lengths of the corresponding sides are proportional. Thus,

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 6 Slide 155Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 7 Slide 156Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. How high is a flagpole that casts a 56- ft shadow at the same time that a 6-ft man casts a 5-ft shadow?

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 7 Slide 157Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. If we use the sun’s rays to represent the third side of the triangle in our drawing of the situation, we see that we have similar triangles. Let p = the height of the flagpole. The ratio of 6 to p is the same as the ratio of 5 to 56. Thus we have the proportion

EXAMPLE 8.8 Similar Triangles b Find lengths of sides of similar triangles using proportions. 7 Slide 158Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.