Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Geometry and Measurement 9 9.1 Systems of Linear Measurement 9.2 Converting Units of Area 9.3 More with Perimeter and Area 9.4 Volume and Capacity 9.5 Angles and Triangles 9.6 Square Roots and the Pythagorean Theorem 9.7 Weight, Mass, and Temperature 9.8 Medical Applications Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 2

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. b Classify an angle as right, straight, acute, or obtuse. c Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. d Classify a triangle as equilateral, isosceles, or scalene, and as right, obtuse, or acute. c Determine whether an ordered pair is a solution of an equation with two variables. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 3

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles e Given two of the angle measures of a triangle, find the third. c Determine whether an ordered pair is a solution of an equation with two variables. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 4

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. The angle above can be named Note that the name of the vertex is either in the middle or, if no confusion results, listed by itself. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 5

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 6

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. To measure angles, we start with some predetermined angle and assign to it a measure of 1. We call it a unit angle. The unit most commonly used for angle measure is the degree. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 7

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. A device called a protractor is used to measure angles. Protractors often have two scales. In the center of the protractor is a vertex indicator such as or a small hole. To measure an angle like on the next slide, 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 below, is on the inside scale, so we check where the angle’s other side crosses the inside scale. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 8

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. We see that m ∠Q = 145°. The notation m ∠Q is read “the measure of angle Q.” Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 9

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. A protractor can be used to draw a circle graph. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 10

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. 1 Transportation. According to a recent poll, 45% of adults believe that flying is the safest mode of transportation, 39% believe that cars are the safest, and 16% believe that trains are the safest. Draw a circle graph to represent these figures. Source: Marist Institute for Public Opinion Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 11

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. 1 Transportation. We begin by drawing a 162 angle. Beginning at the center of the circle, we draw a horizontal segment to the circle. That segment is one side of the angle. We use a protractor to mark off a 162 angle. From that mark, we draw a segment to the center of the circle to complete the angle. This section of the circle graph we label with both the percent (45%) and the type of transportation (Airplanes). Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 12

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. 1 Transportation. From the second segment drawn, we repeat the above procedure to draw a 140.4 angle. Since protractors are marked in units of we must approximate this angle. This section we label with 39% and Cars. The remainder of the circle represents Trains and should be a 57.6 angle; we measure to confirm this, and label the section with 16% and Trains. Finally, we give a title to the graph: Safest Mode of Transportation. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 13

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. 1 Transportation. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 14

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles TYPES OF ANGLES b Classify an angle as right, straight, acute, or obtuse. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 15

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles b Classify an angle as right, straight, acute, or obtuse. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 16

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles b Classify an angle as right, straight, acute, or obtuse. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 17

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles COMPLEMENTARY ANGLES c Two angles are complementary if the sum of their measures is 90 Each angle is called a complement of the other. Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 18

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles c Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. When complementary angles are adjacent to each other (that is, they have a side in common), they form a right angle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 19

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles c Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. 3 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 20

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles SUPPLEMENTARY ANGLES c Two angles are supplementary if the sum of their measures is 180 Each angle is called a supplement of the other. Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 21

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles c Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. Note that when supplementary angles are adjacent, they form a straight angle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 22

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles c Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. 5 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 23

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles c Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. When two lines intersect, four angles are formed. The pairs of angles that do not share any side in common are said to be vertical (or opposite) angles. Thus, in the drawing below, and are vertical angles, as are and Note that m ∠1 = m ∠3 and m ∠4 = m ∠2. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 24

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles VERTICAL ANGLES c Two angles are vertical if they are formed by two intersecting lines and have no side in common. Vertical angles have the same measure. Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 25

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles c Identify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. If two angles have the same measure, we say that they are congruent, denoted by the symbol We do not write that angles are equal: The measures are equal and the angles are congruent. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 26

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles d Classify a triangle as equilateral, isosceles, or scalene, and as right, obtuse, or acute. A triangle is a polygon made up of three segments, or sides. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 27

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles TYPES OF TRIANGLES 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. d Classify a triangle as equilateral, isosceles, or scalene, and as right, obtuse, or acute. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 28

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles e Given two of the angle measures of a triangle, find the third. The sum of the angle measures of every triangle is To see this, note that we can think of cutting apart a triangle as shown on the left below. If we reassemble the pieces, we see that a straight angle is formed. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 29

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles SUM OF THE ANGLE MEASURES OF A TRIANGLE e Given two of the angle measures of a triangle, find the third. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 30

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles e Given two of the angle measures of a triangle, find the third. If we know the measures of two angles of a triangle, we can calculate the measure of the third angle. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 31

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles e Given two of the angle measures of a triangle, find the third. 7 Find the missing angle measure. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 32

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 9.5 Angles and Triangles e Given two of the angle measures of a triangle, find the third. 7 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 33