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

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OBJECTIVES 9.5 Angles and Triangles Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aName an angle in six different ways and measure an angle with a protractor. bClassify an angle as right, straight, acute, or obtuse. cIdentify complementary, supplementary, and vertical angles and find the measure of a complement or a supplement of a given angle. dClassify a triangle as equilateral, isosceles, or scalene, and as right, obtuse, or acute.

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OBJECTIVES 9.5 Angles and Triangles Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. eGiven two of the angle measures of a triangle, find the third.

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9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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.

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

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9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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.

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9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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 protractors at the vertex and line up one of the angles sides at Then we check where the angles other side crosses the scale. In the figure below, is on the inside scale, so we check where the angles other side crosses the inside scale.

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

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

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EXAMPLE 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. 1Transportation. Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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

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EXAMPLE 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. 1Transportation. Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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).

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EXAMPLE 9.5 Angles and Triangles a Name an angle in six different ways and measure an angle with a protractor. 1Transportation. Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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.

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

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9.5 Angles and Triangles TYPES OF ANGLES Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

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

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

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

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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. Slide 19Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. When complementary angles are adjacent to each other (that is, they have a side in common), they form a right angle.

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EXAMPLE 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 Slide 20Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

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

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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. Slide 22Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Note that when supplementary angles are adjacent, they form a straight angle.

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EXAMPLE 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 Slide 23Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

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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. Slide 24Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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.

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9.5 Angles and Triangles VERTICAL ANGLES Slide 25Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Two angles are vertical if they are formed by two intersecting lines and have no side in common. Vertical angles have the same measure.

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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. Slide 26Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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.

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

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9.5 Angles and Triangles TYPES OF TRIANGLES Slide 28Copyright 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.

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9.5 Angles and Triangles e Given two of the angle measures of a triangle, find the third. Slide 29Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 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.

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9.5 Angles and Triangles SUM OF THE ANGLE MEASURES OF A TRIANGLE Slide 30Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

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

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

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

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