Looking Ahead to Chapter 3

Transcription

1 Looking Ahead to Chapter Focus In Chapter, you will learn how to name, measure, and classify angles and triangles. You will also learn about special angles, as well as the triangle inequality. Chapter Warmup Answer these questions to help you review skills that you will need in Chapter. Find each sum Tell whether the statement is true or false Read the problem scenario below. You are building a triangular sandbox. You want the perimeter of the sandbox to be no more than 25 feet. 7. Write an inequality to represent the perimeter of the sandbox. Let a, b, and c represent the side lengths of the sandbox. 8. Suppose that the length of one side of the sandbox is 8 feet and the length of another side of the sandbox is 10 feet. What is the greatest possible length of the third side of the sandbox? Key Terms point p. 114 line p. 114 line segment p. 114 endpoints p. 114 ray p. 114 initial point p. 114 angle p. 115 vertex p. 115 sides p. 115 measure of an angle p. 116 degrees p. 117 straight angle p. 117 right angle p. 117 acute angle p. 118 obtuse angle p. 118 congruent angles p. 120 adjacent angles p. 122 supplementary angles p. 122 complementary angles p. 124 vertical angles p. 126 opposite rays p. 126 linear pair p. 127 interior angle p. 129 theorem p. 10 proof p. 10 exterior angle p. 1 acute triangle p. 19 right triangle p. 19 obtuse triangle p. 140 equiangular triangle p. 140 equilateral triangle p. 141 isosceles triangle p. 141 scalene triangle p. 142 Triangle Inequality p Chapter Introduction to Angles and Triangles

2 CHAPTER Introduction to Angles and Triangles Origami is the art of folding paper. With origami, it is possible to create beautifully intricate designs and shapes while using only a small number of folds. In Lesson.4, you will create an origami bird and then classify the triangles made by the folds..1 Constellations Naming, Measuring, and Classifying Angles p Cable-Stayed Bridges Special Angles p Designing a Kitchen Angles of a Triangle p Origami Classifying Triangles p Building a Shed The Triangle Inequality p. 145 Chapter Introduction to Angles and Triangles 111

3 112 Chapter Introduction to Angles and Triangles

4 .1 Constellations Naming, Measuring, and Classifying Angles Objectives In this lesson, you will: Identify and name lines, line segments, rays, and angles. Measure angles. Classify angles. Key Terms point line line segment endpoints ray initial point angle vertex sides straight angle right angle acute angle obtuse angle measure of an angle degrees congruent angles SCENARIO A constellation is a grouping of stars that form a recognizable pattern. Today there are 88 such groupings. Three constellations you may be familiar with are Ursa Major (which includes the Big Dipper), Ursa Minor (which includes the Little Dipper), and Orion (the Hunter). Problem 1 Orion the Hunter Part of a simplified diagram of the constellation Orion is shown below. A F G H C D B E A. Each star is represented by a point, and each point is named with a letter. We will use our normal alphabet, but typically, the Greek alphabet is used to name the stars. The eight brightest stars in the constellation are labeled. The brightest star in the constellation, point A, is the star Betelgeuse. Use letters to label the rest of the points in the constellation. Do not use a letter more than once. B. Line segments are used to connect the stars in the constellation so that you can better recognize the grouping. Draw a line segment from point A to point H. You can write this line segment symbolically as AH. The symbol AH is read as line segment AH. C. Complete the diagram of Orion above by drawing the line segments listed below. CH, CD, AF, EF, DE, FG, BD D. Do you think that CH is the same as HC? Why or why not? Use a complete sentence in your answer. Lesson.1 Naming, Measuring, and Classifying Angles 11

5 Problem 1 Orion the Hunter E. Can you see why this constellation is called the Hunter? Use complete sentences to explain your reasoning. Investigate Problem 1 1. Just the Math: Rays, Lines, and Line Segments The diagram of Orion is formed from some basic objects in geometry: points and line segments. Other basic objects in geometry are rays and lines. Graphical representations of these objects are shown below. Point: Line: A A B Line segment: A B Ray: A B How are the graphical representations of a line, a line segment, and a ray the same? How are they different? Use complete sentences in your answer. Take Note Points and lines are undefined terms. This means that these objects are not formally defined in mathematical terms, but their meanings are instead given by descriptions. A point has no length, width, or thickness but has a position. The point above is named point A. A line has length but no width or thickness. It extends without end in two directions. You can name a line by identifying two points on the line. The line above is named line AB. In symbols, you can write AB (or BA ). A line segment is a portion of a line between two points called the endpoints. The line segment above is named line segment AB. Symbolically, you can write AB (or BA). A ray is part of a line that consists of an initial point and all points on the line that lie on one side of the initial point. The ray above is named ray AB. In symbols, write AB. 114 Chapter Introduction to Angles and Triangles

6 Investigate Problem 1 Do you think that AB is the same as BA? Why or why not? Use complete sentences in your answer. 2. Use words to name each object. Then use symbols to name each object. Words: Symbols: Q P Words: C Symbols: Words: Symbols: D M N. Plot two points and name them point A and point B. Then draw ray BA. Plot a point, point C, so that point C does not lie on ray BA. Then draw ray BC. You have drawn an angle. An angle is a figure that is formed from two rays that extend from a common point called the vertex. The sides of the angle are the rays. What is the vertex of your angle? Use a complete sentence in your answer. You can name your angle symbolically in three different ways: B, ABC, or CBA. When you use three letters to name an angle, the middle letter is always the vertex. Lesson.1 Naming, Measuring, and Classifying Angles 115

7 Investigate Problem 1 Why might you want to use three letters to name an angle? Use a complete sentence in your answer. 4. Name all of the angles in the figure below. N P M Q L Problem 2 Angles in Orion Consider the diagram of Orion again. I J K L A F H C D M N P O Q R E G B A. Consider IKJ and LAH. How do these angles compare to each other? Use a complete sentence in your answer. B. The measure of an angle indicates the size of an angle. Which angle do you think has a greater measure, IKJ or LAH? Use complete sentences to explain your reasoning. 116 Chapter Introduction to Angles and Triangles

8 Problem 2 Angles in Orion C. Consider AHC. Draw an angle that has a measure that is greater than AHC and draw an angle that has a measure that is less than AHC. Investigate Problem 2 1. The unit of measure for angles is a degree (º). Consider a circle that is divided into 60 equal pieces. The angle of each piece measures one degree (1º). There are 60º around a circle. 60 Draw an angle that measures 180º. Use complete sentences to explain how you found your answer. Draw an angle that measures 90º. Use complete sentences to explain how you found your answer. An angle that has a measure of 180º is called a straight angle and an angle that has a measure of 90º is called a right angle. Lesson.1 Naming, Measuring, and Classifying Angles 117

9 Investigate Problem 2 A right angle is designated by placing a corner, of the angle as shown below., at the vertex A shorthand notation for writing the measure of B is 90 degrees is m B 90. I J K H L A F G C B M N O D R QP E 2. Just the Math: Classifying Angles You can classify angles according to their angle measure. An angle whose measure is less than 90º is an acute angle and an angle whose measure is between 90º and 180º is an obtuse angle. Complete the second column in the table below by classifying each of the angles from Orion given in the first column. Use complete sentences to explain how you found your answer. Angles IKJ AHC Type of angle LAH HCO CON. You can use a tool called a protractor to measure angles. To measure an angle with a protractor, place the center of the protractor at the vertex of the angle and align the bottom of the protractor with one of the sides of the angle as shown below Measuring an acute angle Measuring an obtuse angle 118 Chapter Introduction to Angles and Triangles

10 Investigate Problem 2 Each line on the curved part of the protractor has two measures, one for an acute angle and one for an obtuse angle. If you identify the angle as acute or obtuse before you read the protractor, it is easier to determine which measure you should read. Would you estimate the measure of the angle below to be 40º or 140º? Use a complete sentence to explain your reasoning. 4. Use a protractor to measure the angles in the table in Question 2. Add a column to the end of the table and record your results. Which angle has the greatest measure? Which angle has the least measure? Use a complete sentence in your answer. 5. Compare your measurements to another student s measurements. Are the measurements exactly the same? If not, use complete sentences to explain why this happens. 6. Without measuring the angles, which angle below do you think has a greater measure? Use a complete sentence to explain your reasoning. Use a protractor to measure each angle. What are the measures of the angles? Lesson.1 Naming, Measuring, and Classifying Angles 119

11 Investigate Problem 2 7. When two angles have the same measure, the angles are congruent. Use a protractor to draw two different angles that have a measure of 50º. 8. Three constellations that are triangles are Triangulum, Triangulum Australe, and Octans. Name, measure, and classify each of the angles inside each triangle. M B E A Triangulum C D Triangulum Australe F L Octans N 9. Describe the different kinds of angles and how many of each kind of angle can be in a triangle. Use complete sentences to explain your reasoning. 120 Chapter Introduction to Angles and Triangles

12 .2 Cable-Stayed Bridges Special Angles Objectives In this lesson, you will: Identify pairs of adjacent angles. Identify complementary and supplementary angles. Identify angles that are complementary and supplementary to given angles. Identify vertical angles. Identify linear pairs. Key Terms adjacent angles supplementary angles complementary angles vertical angles opposite rays linear pair SCENARIO You might be familiar with the Golden Gate Bridge in San Francisco, California. The Golden Gate Bridge is a suspension bridge, a bridge in which the roadway is hung from cables that are attached to pillars. In a suspension bridge, the cables form a U-shape. A cable-stayed bridge is another type of bridge in which the roadway is hung from cables. In a cable-stayed bridge, the cables are attached to pillars and the roadway as shown. Suspension bridge Problem 1 Examining Bridge Angles Part of a simple cable-stayed bridge is shown below. A B C D E F G H I J Cable-stayed bridge A. Name all of the angles in the figure. B. Classify all of the angles in the figure. Lesson.2 Special Angles 121

13 A B Problem 1 Examining Bridge Angles C. Name all of the pairs of angles that share a common vertex and side. Also name the common side that each angle pair shares. C D E H I J F G Take Note The word adjacent means next to. These angle pairs are called adjacent angles. Angle CAD and angle EAF are not adjacent. Explain why. Use a complete sentence in your answer. D. Without measuring, can you determine if the measures of the angles in any of the angle pairs in part (C) have a sum of 180º? If so, name the angle pairs and explain your reasoning. Use complete sentences in your answer. Investigate Problem 1 1. Just the Math: Supplementary Angles Two angles are supplementary if the sum of their measures is 180º. The measure of ADC is 120º. What is the measure of ADF? Use complete sentences to explain how you found your answer. The measure of IBJ is 15º. What is the measure of an angle that is supplementary to IBJ? Show all your work and use a complete sentence in your answer. 122 Chapter Introduction to Angles and Triangles

14 Investigate Problem 1 The measure of CEA is 106º. What is the measure of an angle that is supplementary to CEA? Show all your work and use a complete sentence in your answer. The measure of BGJ is 90º. What is the measure of an angle that is supplementary to BGJ? Show all your work and use a complete sentence in your answer. 2. What kind of angle is supplementary to an obtuse angle? Use a complete sentence to explain your reasoning. What kind of angle is supplementary to an acute angle? Use a complete sentence to explain your reasoning. What kind of angle is supplementary to a right angle? Use a complete sentence to explain your reasoning. Lesson.2 Special Angles 12

15 Investigate Problem 1. Just the Math: Complementary Angles Two angles are complementary if the sum of their measures is 90º. Can you tell without measuring whether any of your angle pairs from part (C) are complementary? If so, name the angle pairs. If not, explain how you know that none of the pairs are complementary. Use complete sentences in your answer. 4. Can an obtuse angle be one of the angles in a pair of complementary angles? Use complete sentences to explain your reasoning. 5. Can a right angle have a complement? Use complete sentences to explain your reasoning. 6. What kinds of angles are complements? Explain your reasoning. Use complete sentences in your answer. 7. The measure of CAF is 45º. What is the measure of an angle that is complementary to CAF? 124 Chapter Introduction to Angles and Triangles

16 Problem 2 Angles in a Bridge Consider one side of a cable-stayed bridge and suppose that you could extend some of the cables above the pillar as shown. A B C D E F G A. Find the measures of ADB and FDG. Use a complete sentence in your answer. B. Find the measures of ADC and EDG. Use a complete sentence in your answer. C. Find the measures of BDC and EDF. Use a complete sentence in your answer. D. What do you notice about each pair of angles in part (A) through part (C)? Use a complete sentence in your answer. E. What do the sides of each pair of angles in part (A) through part (C) have in common? Use complete sentences in your answer. Lesson.2 Special Angles 125

17 Investigate Problem 2 1. Just the Math: Vertical Angles The angles you identified in Problem 2 are vertical angles. Vertical angles are angles whose sides form two pairs of opposite rays. Opposite rays are two rays on the same line that have the same initial point and extend in opposite directions. For each pair of angles in part (A) through part (C), name the pairs of opposite rays. 2. Explain how you would draw a pair of vertical angles. Use complete sentences in your answer.. What can you conclude about the measures of vertical angles? Use a complete sentence in your answer. 4. Are there any other vertical angles in the diagram in Problem 2? If so, name the angles and the pairs of opposite rays. 5. Determine whether each pair of given angles are vertical angles. If not, explain why. Use a complete sentence in your answer. L M N S T R LSM and PSR P Q 126 Chapter Introduction to Angles and Triangles

18 Investigate Problem 2 MSN and PSQ LSM and MSN LSN and TSP LSM and PSQ 6. Describe everything that you know about LSQ and QSP. Use complete sentences in your answer. Describe everything that you know about LSM and MSP. Use complete sentences in your answer. Each pair of angles in Question 6 forms a linear pair. Two adjacent angles form a linear pair if the noncommon sides of the angles are opposite rays. 7. Do NSP and PSQ form a linear pair? Why or why not? Use a complete sentence in your answer. 8. List all of the linear pairs of angles given in the figure in Question Complete the following statement: The angles in a linear pair are angles because the sum of their measures is. Lesson.2 Special Angles 127

19 Investigate Problem Determine whether the following statements are true or false. If a statement is true, explain your reasoning. If a statement is false, give an example that shows that the statement is false. Use complete sentences in your answer. All supplementary angles form a linear pair. All vertical angles are nonadjacent. All adjacent angles are either complementary or supplementary. Two complementary angles cannot be vertical angles. 128 Chapter Introduction to Angles and Triangles

20 . Designing a Kitchen Angles of a Triangle Objectives In this lesson, you will: Name triangles. Find sums of measures of angles in triangles. Use the Triangle Sum Theorem. Use the Exterior Angle Theorem. Discover the Exterior Angle Inequality. Key Terms interior angle theorem proof exterior angle SCENARIO Before the layout of a kitchen is finalized, particular attention is paid to where the sink, stove, and refrigerator are placed. This is so that someone working in the kitchen can prepare food efficiently. The work triangle is the triangle formed by the locations of the sink, stove, and refrigerator. The work triangle should have a perimeter between 12 feet and 26 feet. A larger or smaller perimeter would result in an inefficient layout. Problem 1 Kitchen Layout A common kitchen design is a galley kitchen. In a galley kitchen, the cabinets and appliances are placed on opposite sides of the room from each other. A. A galley kitchen is shown below. The stove is labeled with an S, the sink is labeled with a K, and the refrigerator is labeled with an R. Draw the work triangle. K R S B. You can name a triangle by using the names of the vertices. For instance, you can name the work triangle in part (A) triangle SKR. Symbolically, you can write SKR. Use symbols to list the other ways that you can name the work triangle. C. What geometric description can you give to the sides of a triangle? Use a complete sentence in your answer. Then name the sides of SKR. D. How many angles are inside SKR? Name the angles. These angles are called the interior angles of the triangle. Lesson. Angles of a Triangle 129

21 Problem 1 Kitchen Layout E. Trace SKR from part (A) onto a sheet of paper and cut out the triangle. Then tear off the three angles of the triangle. Then place the angles next to each other so that one side of an angle coincides with the side of another angle. What do you notice about the sum of these angles? Use a complete sentence to explain your reasoning. F. Two other kitchen designs are L-shaped kitchens and U-shaped kitchens. Each of these kitchens is shown below. Draw the work triangle for each kitchen. S S K R K R Now repeat the process in part (E) for each of these work triangles. What do you notice about the sum of these angles? Use a complete sentence to explain your reasoning. Investigate Problem 1 1. What do you think is the sum of the interior angles of any triangle? Use a complete sentence in your answer. We can write a theorem that states this result. A theorem is a statement that has been shown, or proven, to be true. The process of showing that a statement is true is called a proof. Triangle Sum Theorem The sum of the interior angles of a triangle is 180º. Your work above in Problem 1 is an informal proof. You will learn about formal proofs later in Chapter Chapter Introduction to Angles and Triangles

22 Investigate Problem 1 2. Classify the interior angles in your work triangles in Problem 1. Use complete sentences in your answer.. Is it possible to draw a triangle that has three acute interior angles? If so, draw the triangle. If not, use complete sentences to explain why not. 4. Is it possible to draw a triangle that has exactly two acute interior angles? If so, draw the triangle. If not, use complete sentences to explain why not. 5. Is it possible to draw a triangle that has exactly one right interior angle? If so, draw the triangle. If not, use complete sentences to explain why not. 6. Is it possible to draw a triangle that has exactly one acute interior angle? If so, draw the triangle. If not, use complete sentences to explain why not. Lesson. Angles of a Triangle 11

23 Investigate Problem 1 7. Is it possible to draw a triangle that has exactly two right interior angles? If so, draw the triangle. If not, use complete sentences to explain why not. 8. Is it possible to draw a triangle that has exactly one obtuse interior angle? If so, draw the triangle. If not, use complete sentences to explain why not. 9. Is it possible to draw a triangle that has exactly two obtuse interior angles? If so, draw the triangle. If not, use complete sentences to explain why not. 10. Is it possible to draw a triangle that has one obtuse interior angle and one right interior angle? If so, draw the triangle. If not, use complete sentences to explain why not. 11. What property of a triangle helped you answer Question through Question 10? 12. Describe the combinations of types of angles that are possible for the interior angles of a triangle. Use complete sentences in your answer. 12 Chapter Introduction to Angles and Triangles

24 Problem 2 Extending the Sides A. There are also exterior angles associated with triangles. An exterior angle is formed when one of the sides of a triangle is extended into a ray. An exterior angle of a triangle is adjacent to an interior angle. Name the exterior angle drawn on ABC below. Use a complete sentence in your answer. B A C D B. How many exterior angles do you think a triangle has? Use complete sentences to explain your reasoning. C. What is the sum of the measures of BCA and BCD? Explain your reasoning. Use a complete sentence in your answer. D. What is the sum of the measures of A, B, and BCA? Use a complete sentence in your answer. E. How do the sums of the measures in part (C) and part (D) compare? Use a complete sentence in your answer. F. What angle s measure do these sums have in common? Use a complete sentence in your answer. What is the relationship between the remaining angles? Use a complete sentence in your answer. Lesson. Angles of a Triangle 1

25 Take Note Another way to name angles is to use positive integers, as shown at the right. Also, the symbol for the measure of is m. For instance, the measure of 1 can be written as m 1. Investigate Problem 2 1. Just the Math: Exterior Angle Theorem The result of Problem 2 is called the Exterior Angle Theorem. Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. 1 2 m 1 + m 2 = m 4 4 Write the possible sums of angle measures given by the Exterior Angle Theorem for the triangle below Consider the galley kitchen work triangle from Problem 1. K R 1 S Tell what you know about the measure of 1 in terms of the measures of the interior angles of the triangle without finding the measures of the angles. Justify your reasoning. Use complete sentences in your answer. How does the measure of 1 compare to the measure of S? Use a complete sentence to explain your reasoning. 14 Chapter Introduction to Angles and Triangles

26 Investigate Problem 2 How does the measure of 1 compare to the measure of R? Use a complete sentence to explain your reasoning. The relationship between the measures of an exterior angle and each of the nonadjacent interior angles of a triangle can be formally stated as a theorem. Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent interior angles. 2 m 4 > m 1 m 4 > m For each triangle below, find the measure of 1. Show all your work. State the theorem you used to find your answer Lesson. Angles of a Triangle 15

27 16 Chapter Introduction to Angles and Triangles

28 .4 Origami Classifying Triangles Objectives In this lesson, you will: Classify triangles by angles. Classify triangles by side lengths. Key Terms acute triangle right triangle obtuse triangle equiangular triangle equilateral triangle isosceles triangle scalene triangle SCENARIO Origami is the art of paper folding. You can use a single sheet of paper and fold it many times in different ways to create origami animals, boxes, hats, and many other objects. Problem 1 Create an Origami Bird Use the instructions below to create an origami bird. Start with a square piece of paper. A. Fold the paper in half along the diagonal as shown. Then open the paper back up. B. Fold both sides into the diagonal. C. Fold the right half of the paper over the diagonal. D. Fold up the top flap of paper along the dotted line. Then turn the paper over and fold the other flap of paper up and flip the paper back over. Lesson.4 Classifying Triangles 17

29 Problem 1 Create an Origami Bird E. Fold the top vertex of the triangle back to form the bird s neck. F. Fold the top vertex forward to form the bird s head. G. Fold the bottom vertex up to form the bird s tail. Investigate Problem 1 1. If you unfolded your bird, you would see the folds that were made to create your bird, as shown below. A B C M L R N P D K S Q T J H G F E 18 Chapter Introduction to Angles and Triangles

30 Investigate Problem 1 Name all of the triangles formed by the folds. How many triangles do you see being formed by the folds? Use a complete sentence in your answer. 2. Consider GQF. Classify the interior angles of this triangle. Use a complete sentence in your answer. This triangle can be classified by its interior angles as an acute triangle. Why? Use a complete sentence in your answer.. Consider GCE. Classify the interior angles of this triangle. This triangle can be classified by its interior angles as a right triangle. Why? Use a complete sentence in your answer. How many angles in a right triangle are not right angles? Classify these angles. Use a complete sentence in your answer. What is the sum of the measures of these two angles? How do you know? Use complete sentences in your answer. Will the sum of the measures of the two nonright angles always be the same in any right triangle? Why or why not? Use a complete sentence in your answer. Lesson.4 Classifying Triangles 19

31 A M L K N R S T B P Q C D Investigate Problem 1 4. Consider NCP. Classify the interior angles of this triangle. This triangle can be classified by its interior angles as an obtuse triangle. Why? Use a complete sentence in your answer. J H G F E How many angles in an obtuse triangle are not obtuse angles? Classify these angles. Use a complete sentence in your answer. Can you find the sum of the measures of these angles without using a protractor? Why or why not? Use a complete sentence in your answer. Will the sum of the measures of these angles always be the same in any obtuse triangle? Why or why not? Use a complete sentence in your answer. 5. An equiangular triangle is a triangle in which all three interior angles have the same measure. What is the measure of an interior angle of an equiangular triangle? Use a complete sentence to explain your reasoning. 6. Classify each of the triangles in Question 1 according to its angle measures. 140 Chapter Introduction to Angles and Triangles

32 Investigate Problem 1 7. Draw an equiangular triangle. Then measure the lengths of the sides of your triangle in centimeters. Round your answer to the nearest tenth of a centimeter. What do you notice about the lengths of the sides of the triangle? Use a complete sentence in your answer. 8. A triangle in which the sides all have the same length is called an equilateral triangle. Draw an equilateral triangle. Then measure the interior angles of the triangle. What do you notice about the measures of the angles in an equilateral triangle? Use a complete sentence in your answer. 9. What can you conclude about all equiangular triangles? What can you conclude about all equilateral triangles? Use complete sentences in your answer. 10. Name the equilateral triangles from the triangles listed in Question An isosceles triangle is a triangle in which at least two of the sides have the same length. Identify the triangles from Question 1 that appear to be isosceles. Lesson.4 Classifying Triangles 141

33 Investigate Problem 1 Measure the interior angles of the isosceles triangles. You do not have to list the measures. What can you conclude about the angle measures of isosceles triangles? Use a complete sentence in your answer. 12. A scalene triangle is a triangle in which none of the sides have the same length. Name the scalene triangles from the triangles listed in Question 1. Measure the interior angles of 5 scalene triangles picked at random. You do not have to list the measures. What can you conclude about the angle measures of scalene triangles? Use a complete sentence in your answer. 1. Is it possible to draw an isosceles right triangle? If so, draw the triangle. If not, use complete sentences to explain why not. 14. Is it possible to draw an isosceles obtuse triangle? If so, draw the triangle. If not, use complete sentences to explain why not. 15. Is it possible to draw a scalene obtuse triangle? If so, draw the triangle. If not, use complete sentences to explain why not. 142 Chapter Introduction to Angles and Triangles

34 Investigate Problem Is it possible to draw a scalene right triangle? If so, draw the triangle. If not, use complete sentences to explain why not. 17. Is it possible to draw a scalene acute triangle? If so, draw the triangle. If not, use complete sentences to explain why not. Lesson.4 Classifying Triangles 14

35 144 Chapter Introduction to Angles and Triangles

36 .5 Building a Shed The Triangle Inequality Objectives In this lesson, you will: Use the Triangle Inequality. Investigate the relationship between the sides and angles of triangles. SCENARIO A person buys a kit that contains the parts needed to build the storage shed shown below. The kit contains 12 pieces of pre-cut wood that are used for the frame of the shed. The kit s instructions do not give directions on which lengths of wood to use for the roof and which lengths of wood to use for building the main part of the frame. Key Term Triangle Inequality Problem 1 Spaghetti Experiment In this experiment, you will try to form triangles from pieces of pasta. A. Choose three pieces of the pasta that your teacher gave you and measure each of the pieces in centimeters. Record the lengths in the table below. Then determine whether the pieces of pasta can be formed into a triangle. Record the result in the table. Run Piece 1 (centimeters) Piece 2 (centimeters) Piece (centimeters) Triangle? (yes/no) Lesson.5 The Triangle Inequality 145

37 Problem 1 Spaghetti Experiment B. Run the experiment as many times as you can so that none of the combinations of side lengths are repeated in the table. Record all of your results in the table on the previous page. C. Consider the pieces of pasta that did not form triangles. Did you notice anything about the lengths of the pieces as they related to one another when you were trying to form the triangle? Use complete sentences to describe what you discovered. D. Consider the pieces of pasta that did form triangles. Did you notice anything about the lengths of the pieces as they related to one another when you were trying to form the triangle? Use complete sentences to describe what you discovered. E. Suppose that you have three more pieces of pasta with lengths of centimeters, 4 centimeters, and 8 centimeters. Can these pieces of pasta form a triangle? Use complete sentences to explain your reasoning. Suppose that you have three more pieces of pasta with lengths of 8 centimeters, 12 centimeters, and 15 centimeters. Can these pieces of pasta form a triangle? Use complete sentences to explain your reasoning. Investigate Problem 1 1. Just the Math: The Triangle Inequality In Problem 1, you discovered a rule called the Triangle Inequality. Triangle Inequality The length of one side of a triangle is less than the sum of the lengths of the other sides of the triangle. 146 Chapter Introduction to Angles and Triangles

38 Take Note To symbolically write the length of segment AB, you write AB. Investigate Problem 1 Symbolically, if A, B, and C are vertices of a triangle, then AC AB BC. C A B What other ways can you symbolically write the Triangle Inequality? 2. The frame of the storage shed is formed from four pieces of wood that are 45 inches long, four pieces of wood that are 54 inches long, and four pieces of wood that are 114 inches long. The frame on the sides of the shed forms four rectangles with the base, and the frame on the top of the shed forms four triangles. Two of the sides of each triangle have the same length. Draw and label diagrams that show the possible dimensions of the triangles that may form the roof of the shed.. Which of the triangles in Question 2 could be used to form the roof? Explain how you know. Use complete sentences in your answer. Lesson.5 The Triangle Inequality 147

39 Investigate Problem 1 4. Suppose that you know that the longest side of a roof triangle is the one that sits on top of the frame for the sides of the shed. Label the shed diagram with the correct dimensions. Problem 2 Roof Angles and Sides An accurate model of the roof shape is shown below. B A C A. Find the lengths of the sides of the model. Round your answers to the nearest tenth of a centimeter. Label the sides of the model with these lengths. B. Find the measures of the interior angles of the model. Label the interior angles with their measures. C. Which side(s) have the greatest length? Which angle(s) have the greatest measure? Use a complete sentence in your answer. What is the relationship between the side(s) with the greatest length and the angle(s) with the greatest measure? Use a complete sentence in your answer. 148 Chapter Introduction to Angles and Triangles

40 Problem 2 Roof Angles and Sides D. Which side(s) have the least length? Which angle(s) have the least measure? Use a complete sentence in your answer. What is the relationship between the side(s) with the least length and the angle(s) with the least measure? Use a complete sentence in your answer. Investigate Problem 2 1. Do you think that the relationships in part (C) and part (D) are true for any triangle? Use complete sentences to explain your reasoning. 2. List the angles in order of least to greatest measurements. List the sides in order of least to greatest measurements. Do not measure the angles or the sides. D E F 45 millimeters L 64 millimeters M 29 millimeters N Lesson.5 The Triangle Inequality 149