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to use high-speed computers to create the figure below, called the Mandelbrot set. Only the black area is part of the set itself. The rainbow colors represent properties of points near the Mandelbrot set. To learn more about different kinds of fractals, visit www.keymath.com/DG. EXPLORATION Patterns in Fractals 137 ● CHAPTER 11 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER CHAPTER 2 R E V I E W This chapter introduced you to inductive reasoning. You used inductive reasoning to observe patterns and make conjectures. You learned to disprove a conjecture with a counterexample and to explain why a conjecture is true with deductive reasoning. You learned how to predict number sequences with rules and how to use these rules to model application problems. Then you discovered special relationships about angle pairs and made your first geometry conjectures. Finally you explored the properties of corresponding, alternate interior, and alternate exterior angles formed by a transversal across parallel lines. As you review the chapter, be sure you understand all the important terms. Go back to the lesson to review any terms you’re unsure of. EXERCISES 1. “My dad is in the navy, and he says that food is great on submarines,” said Diana. “My mom is a pilot,” added Jill, “and she says that airline food is notoriously bad.” “My mom is an astronaut trainee,” said Julio, “and she says that astronauts’ food is the worst imaginable.” “You know,” concluded Diana, “I bet no life exists beyond Earth! As you move farther and farther from the surface of Earth, food tastes worse and worse. At extreme altitudes, food must taste so bad that no creature could stand to eat. Therefore, no life exists out there.” What do you think of Diana’s reasoning? Is it inductive or deductive? 2. Think of a situation you observed outside of school in which inductive reasoning was used incorrectly. Write a paragraph or two describing what happened and explaining why you think it was poor inductive reasoning. 3. Think of a situation you observed outside of school in which deductive reasoning was used incorrectly. Write a paragraph or two describing what happened and explaining why you think it was poor deductive reasoning. For Exercises 4 |
–7, find the next two terms in the sequence. 4. 7, 21, 35, 49, 63, 77,?,? 6. 7, 2, 5, 3, 8, 11,?,? 5. Z, 1, Y, 2, X, 4, W, 8,?,? 7. A, 4, D, 9, H, 16, M, 25,?,? For Exercises 8 and 9, generate the first six terms in the sequence for each function rule. 8. f(n) n2 1 9. f(n) 2n1 For Exercises 10 and 11, draw the next shape in the pattern. 10. 11. 138 CHAPTER 2 Reasoning in Geometry ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER 2 For Exercises 12–15, find the nth term and the 20th term in the sequence. 12. 13. 14. 15. n f(n) n f(n) n f(n) n f(n 10 13 … 16 4 6 4 10 5 25 5 10 5 15 6 … 36 … 6 … 15 … 6 … 21 … n n n n … 20 … … 20 … … 20 … … 20 … For Exercises 16 and 17, find a relationship. Then complete the conjecture. 16. Conjecture: The sum of the first 30 odd whole numbers is?. 17. Conjecture: The sum of the first 30 even whole numbers is?. 18. Viktoriya is a store window designer for Savant Toys. She plans to build a stack of blocks similar to the ones shown below but 30 blocks high. Make a conjecture for the value of the nth term and for the value of the 30th term. How many blocks will she need? 19. The stack of bricks at right is four bricks high. Find the total number of bricks for a stack that is 100 bricks high. 20. For the 4-by-7 rectangular grid, the diagonal passes through 10 squares and 9 interior segments. In an 11-by-101 grid of squares, how many squares will the diagonal pass through? How many interior segments will it pass through? 21. If at a party there are a total of 741 handshakes and each person shakes hands with everyone else at the party exactly once, how many people are at the party? 22. |
If 28 lines are drawn on a plane, what is the maximum number of points of intersection possible? 23. If a whole bunch of lines (no two parallel, no three concurrent) intersect in a plane 2926 times, how many lines are a whole bunch? CHAPTER 2 REVIEW 139 EW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTE 24. If in a 54-sided polygon all possible diagonals are drawn from one vertex, they divide the interior of the polygon into how many regions? 25. How many sides does the polygon have if all possible diagonals drawn from one vertex divide the interior of the polygon into 54 regions? d 26. Trace the diagram at right. Calculate each lettered angle measure. 142° a c b f e 130° g h 106° Assessing What You’ve Learned WRITE IN YOUR JOURNAL Many students find it useful to reflect on the mathematics they’re learning by keeping a journal. Like a diary or a travel journal, a mathematics journal is a chance for you to reflect on what happens each day and your feelings about it. Unlike a diary, though, your mathematics journal isn’t private—your teacher may ask to read it too, and may respond to you in writing. Reflecting on your learning experiences will help you assess your strengths and weaknesses, your preferences, and your learning style. Reading through your journal may help you see what obstacles you have overcome. Or it may help you realize when you need help. So far, you have written definitions, looked for patterns, and made conjectures. How does this way of doing mathematics compare to the way you have learned mathematics in the past? What are some of the most significant concepts or skills you’ve learned so far? Why are they significant to you? What are you looking forward to in your study of geometry? What are your goals for this class? What specific steps can you take to achieve your goals? What are you uncomfortable or concerned about? What are some things you or your teacher can do to help you overcome these obstacles? KEEPING A NOTEBOOK You should now have four parts to your notebook: a section for homework and notes, an investigation section, a definition list, and now a conjecture list. Make sure these are up-to-date. UPDATE YOUR PORTFOLIO Choose one or more pieces of your most significant work in this chapter |
to add to your portfolio. These could include an investigation, a project, or a complex homework exercise. Make sure your work is complete. Describe why you chose the piece and what you learned from it. 140 CHAPTER 2 Reasoning in Geometry CHAPTER 3 Using Tools of Geometry There is indeed great satisfaction in acquiring skill, in coming to thoroughly understand the qualities of the material at hand and in learning to use the instruments we have—in the first place, our hands!—in an effective and controlled way. M. C. ESCHER Drawing Hands, M. C. Escher, 1948 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● learn about the history of geometric constructions ● develop skills using a compass, a straightedge, patty paper, and geometry software ● see how to create complex figures using only a compass, a straightedge, and patty paper L E S S O N 3.1 It is only the first step that is difficult. MARIE DE VICHY-CHAMROD Duplicating Segments and Angles The compass, like the straightedge, has been a useful geometry tool for thousands of years. The ancient Egyptians used the compass to mark off distances. During the Golden Age of Greece, Greek mathematicians made a game of geometric constructions. In his work Elements, Euclid (325–265 B.C.E.) established the basic rules for constructions using only a compass and a straightedge. In this course you will learn how to construct geometric figures using these tools as well as patty paper. Constructions with patty paper are a variation on the ancient Greek game of geometric constructions. Almost all the figures that can be constructed with a compass and a straightedge can also be constructed using a straightedge and patty paper, waxed paper, or tracing paper. If you have access to a computer with a geometry software program, you can do constructions electronically. In the previous chapters, you drew and sketched many figures. In this chapter, however, you’ll construct geometric figures. The words sketch, draw, and construct have specific meanings in geometry. Mathematics Euclidean geometry is the study of geometry based on the assumptions of Euclid (325–265 B.C.E.). Euclid established the basic rules for constructions using only a compass and a straightedge. In his work Elements, Euclid proposed definitions and constructions about points, lines |
, angles, surfaces, and solids. He also explained why the constructions were correct with deductive reasoning. A page from a book on Euclid, above, shows some of his constructions and a translation of his explanations from Greek into Latin. When you sketch an equilateral triangle, you may make a freehand sketch of a triangle that looks equilateral. You don’t need to use any geometry tools. When you draw an equilateral triangle, you should draw it carefully and accurately, using your geometry tools. You may use a protractor to measure angles and a ruler to measure the sides to make sure they are equal in measure. 142 CHAPTER 3 Using Tools of Geometry When you construct an equilateral triangle with a compass and straightedge, you don’t rely on measurements from a protractor or ruler. You must use only a compass and a straightedge. This method of construction guarantees that your triangle is equilateral. When you construct an equilateral triangle with patty paper and straightedge, you fold the paper and trace equal segments. You may use a straightedge to draw a segment, but you may not use a compass or any measuring tools. When you sketch or draw, use the special marks that indicate right angles, parallel segments, and congruent segments and angles. By tradition, neither a ruler nor a protractor is ever used to perform geometric constructions. Rulers and protractors are measuring tools, not construction tools. You may use a ruler as a straightedge in constructions, provided you do not use its marks for measuring. In the next two investigations you will discover how to duplicate a line segment using only your compass and straightedge, or using only patty paper. Investigation 1 Copying a Segment You will need ● a compass ● a straightedge ● a ruler ● patty paper B A C A B C Stage 1 Stage 2 D Stage 3 Step 1 Step 2 Step 3 The complete construction for copying a segment, AB, is shown above. Describe each stage of the process. Use a ruler to measure AB and CD. How do the two segments compare? Describe how to duplicate a segment using patty paper instead of a compass. LESSON 3.1 Duplicating Segments and Angles 143 Using only a compass and a straightedge, how would you duplicate an angle? In other words, how would you construct an angle that is congruent to a given angle? You may not use your protractor, because a protractor |
is a measuring tool, not a construction tool. You will need ● a compass ● a straightedge Investigation 2 Copying an Angle D E F Step 1 The first two stages for copying DEF are shown below. Describe each stage of the process. D D E F G Stage 1 E F G Stage 2 Step 2 Step 3 What will be the final stage of the construction? Use a protractor to measure DEF and G. What can you state about these angles? Step 4 Describe how to duplicate an angle using patty paper instead of a compass. You’ve just discovered how to duplicate segments and angles using a straightedge and compass or patty paper. These are the basic constructions. You will use combinations of these to do many other constructions. You may be surprised that you can construct figures more precisely without using a ruler or protractor! Called vintas, these canoes with brightly patterned sails are used for fishing in Zamboanga, Philippines. What angles and segments are duplicated in this photo? 144 CHAPTER 3 Using Tools of Geometry EXERCISES Construction Now that you can duplicate line segments and angles, do the constructions in Exercises 1–10. You will duplicate polygons in Exercises 7 and 10. 1. Using only a compass and a straightedge, duplicate the three line segments shown below. Label them as they’re labeled in the figure. You will need Construction tools for Exercises 1–10 Geometry software for Exercise 11 A B C D E F 2. Use the segments from Exercise 1 to construct a line segment with length AB CD. 3. Use the segments from Exercise 1 to construct a line segment with length AB 2EF CD. 4. Use a compass and a straightedge to duplicate each angle. There’s an arc in each angle to help you. 5. Draw an obtuse angle. Label it LGE, then duplicate it. 6. Draw two acute angles on your paper. Construct a third angle with a measure equal to the sum of the measures of the first two angles. Remember, you cannot use a protractor—use a compass and a straightedge only. 7. Draw a large acute triangle on the top half of your paper. Duplicate it on the bottom half, using your compass and straightedge. Do not erase your construction marks, so others can see your method. 8. Construct an equilateral triangle. Each side should be the length of this segment. 9. Repeat Exerc |
ises 7 and 8 using constructions with patty paper. 10. Draw quadrilateral QUAD. Duplicate it, using your compass and straightedge. Label the construction COPY so that QUAD COPY. 11. Technology Use geometry software to construct an equilateral triangle. Drag each vertex to make sure it remains equilateral. LESSON 3.1 Duplicating Segments and Angles 145 Review 12. Copy the diagram at right. Use the Vertical Angles Conjecture and the Parallel Lines Conjecture to calculate the measure of each angle. 130 115° k 25° l 13. Hyacinth is standing on the curb waiting to cross 24th Street. A half block to her left is Avenue J, and Avenue K is a half block to her right. Numbered streets run parallel to one another and are all perpendicular to lettered avenues. If Avenue P is the northernmost avenue, which direction (N, S, E, or W) is she facing? 14. Write a new definition for an isosceles triangle, based on the triangle’s reflectional symmetry. Does your definition apply to equilateral triangles? Explain. 15. Draw DAY after it is rotated 90° clockwise about the origin. Label the coordinates of the vertices. 16. Sketch the three-dimensional figure formed by folding this net into a solid. y 6 Y –6 D A x 6 –6 IMPROVING YOUR ALGEBRA SKILLS Pyramid Puzzle II Place four different numbers in the bubbles at the vertices of each pyramid so that the two numbers at the ends of each edge add to the number on that edge. a 88 105 b 129 d 102 119 c 146 CHAPTER 3 Using Tools of Geometry L E S S O N 3.2 To be successful, the first thing to do is to fall in love with your work. SISTER MARY LAURETTA You will need ● patty paper Constructing Perpendicular Bisectors Each segment has exactly one midpoint. A segment bisector is a line, ray, or segment in a plane that passes through the midpoint of a segment in a plane. B Segment AB has a midpoint O. O A A O n B m Lines, m, and n bisect AB. j O A k B Lines j, k, and are perpendicular to AB. A segment has many perpendiculars and many bisectors, but each segment in a plane has only one bisector that is also perpendicular to |
the segment. This line is its perpendicular bisector. Line is the perpendicular bisector of AB. A O B Investigation 1 Finding the Right Bisector In this investigation you will discover how to construct the perpendicular bisector of a segment. Q P Q Q P P Step 1 Step 2 Step 3 Step 1 Step 2 Step 3 Draw a segment on patty paper. Label it PQ. Fold your patty paper so that endpoints P and Q land exactly on top of each other, that is, they coincide. Crease your paper along the fold. Unfold your paper. Draw a line in the crease. What is the relationship of this line to PQ? Check with others in your group. Use your ruler and protractor to verify your observations. LESSON 3.2 Constructing Perpendicular Bisectors 147 How would you describe the relationship of the points on the perpendicular bisector to the endpoints of the bisected segment? There’s one more step in your investigation. Step 4 Place three points on your perpendicular bisector. Label them A, B, and C. With your compass, compare the distances PA and QA. Compare the distances PB and QB. Compare the distances PC and QC. What do you notice about the two distances from each point on the perpendicular bisector to the endpoints of the segment? Compare your results with the results of others. Then copy and complete the conjecture. A B P Q C Remember to add each conjecture to your conjecture list and draw a figure for it. Perpendicular Bisector Conjecture If a point is on the perpendicular bisector of a segment, then it is? from the endpoints. C-5 You’ve just completed the Perpendicular Bisector Conjecture. What about the converse of this statement? Investigation 2 Right Down the Middle If a point is equidistant, or the same distance, from two endpoints of a line segment in a plane, will it be on the segment’s perpendicular bisector? If so, then locating two such points can help you construct the perpendicular bisector. You will need ● a compass ● a straightedge Step 1 Step 2 Step 1 Step 2 Step 3 Draw a line segment. Using one endpoint as center, swing an arc on one side of the segment. Using the same compass setting, but using the other endpoint as center, swing a second arc intersecting the first. The point where the two arcs intersect is equidistant from the endpoints of your |
segment. Use your compass to find another such point. Use these points to construct a line. Is this line the perpendicular bisector of the segment? Use the paper-folding technique of Investigation 1 to check. 148 CHAPTER 3 Using Tools of Geometry Step 4 Complete the conjecture below, and write a summary of what you did in this investigation. Converse of the Perpendicular Bisector Conjecture If a point is equidistant from the endpoints of a segment, then it is on the? of the segment. C-6 Notice that constructing the perpendicular bisector also locates the midpoint of a segment. Now that you know how to construct the perpendicular bisector and the midpoint, you can construct rectangles, squares, and right triangles. You can also construct two special segments in any triangle: medians and midsegments. The segment connecting the vertex of a triangle to the midpoint of its opposite side is a median. There are three midpoints and three vertices in every triangle, so every triangle has three medians. Median The segment that connects the midpoints of two sides of a triangle is a midsegment. A triangle has three sides, each with its own midpoint, so there are three midsegments in every triangle. EXERCISES Construction For Exercises 1–5, construct the figures using only a compass and a straightedge. 1. Construct and label AB. Construct the perpendicular bisector of AB. 2. Construct and label QD. Construct perpendicular bisectors to divide QD into four congruent segments. Midsegment You will need Construction tools for Exercises 1–10 and 14 Geometry software for Exercise 13 3. Construct a line segment so close to the edge of your paper that you can swing arcs on only one side of the segment. Then construct the perpendicular bisector of the segment. 4. Using AB and CD, construct a segment with length 2AB 1 CD. 2 A B C D 5. Construct MN with length equal to the average length of AB and CD above. LESSON 3.2 Constructing Perpendicular Bisectors 149 6. Construction Do Exercises 1–5 using patty paper. Construction For Exercises 7–10, you have your choice of construction tools. Use either a compass and a straightedge or patty paper and a straightedge. Do not use patty paper and compass together. 7. Construct ALI. Construct the perpendicular bisector of each |
side. What do you notice about the three bisectors? 8. Construct ABC. Construct medians AM, BN, and CL. Notice anything special? 9. Construct DEF. Construct midsegment GH where G is the midpoint of DF and H is the midpoint of DE. What do you notice about the relationship between EF and GH? 10. Copy rectangle DSOE onto your paper. Construct the midpoint of each side. Label the midpoint of DS point I, the midpoint of SO point C, the midpoint of OE point V, and the midpoint of ED point R. Construct quadrilateral RICV. Describe RICV. E D O S 11. The island shown at right has two post offices. The postal service wants to divide the island into two zones so that anyone within each zone is always closer to their own post office than to the other one. Copy the island and the locations of the post offices and locate the dividing line between the two zones. Explain how you know this dividing line solves the problem. Or, pick several points in each zone and make sure they are closer to that zone’s post office than they are to the other one. 12. Copy parallelogram FLAT onto your paper. Construct the perpendicular bisector of each side. What do you notice about the quadrilateral formed by the four lines? F L Review T A 13. Technology Use geometry software to construct a triangle. Construct a median. Are the two triangles created by the median congruent? Use an area measuring tool in your software program to find the areas of the two triangles. How do they compare? If you made the original triangle from heavy cardboard, and you wanted to balance that cardboard triangle on the edge of a ruler, what would you do? 14. Construction Construct a very large triangle on a piece of cardboard or mat board and construct its median. Cut out the triangle and see if you can balance it on the edge of a ruler. Sketch how you placed the triangle on the ruler. Cut the triangle into two pieces along the median and weigh the two pieces. Are they the same weight? 150 CHAPTER 3 Using Tools of Geometry In Exercises 15–20, match the term with its figure below. 15. Scalene acute triangle 16. Isosceles obtuse triangle 17. Isosceles right triangle 18. Isosceles acute triangle 19. Scalene obtuse triangle 20. Scalene right triangle A |
. 50° 8 65° 8 65° D. 6 103° 8 11 B. E. 4 6 4 6 115° C. 3 5 4 F. 7 59° 73° 9 8 48° 21. Sketch and label a polygon that has exactly three sides of equal length and exactly two angles of equal measure. 22. Sketch two triangles. Each should have one side measuring 5 cm and one side measuring 9 cm, but they should not be congruent. 23. List the letters from the alphabet below that have a horizontal line of symmetry IMPROVING YOUR VISUAL THINKING SKILLS Folding Cubes I In the problems below, the figure at the left represents the net for a cube. When the net is folded, which cube at the right will it become? 1. 2. 3. A. A. A. B. B. B. C. C. C. LESSON 3.2 Constructing Perpendicular Bisectors 151 L E S S O N 3.3 Intelligence plus character— that is the goal of true education. MARTIN LUTHER KING, JR. Constructing Perpendiculars to a Line If you are in a room, look over at one of the walls. What is the distance from where you are to that wall? How would you measure that distance? There are a lot of distances from where you are to the wall, but in geometry when we speak of a distance from a point to a line we mean a particular distance. The construction of a perpendicular from a point to a line (with the point not on the line) is another of Euclid’s constructions, and it has practical applications in many fields including agriculture and engineering. For example, think of a high-speed Internet cable as a line and a building as a point not on the line. Suppose you wanted to connect the building to the Internet cable using the shortest possible length of connecting wire. How can you find out how much wire you need, without buying too much? Investigation 1 Finding the Right Line You already know how to construct perpendicular bisectors. You can’t bisect a line because it is infinitely long, but you can use that know-how to construct a perpendicular from a point to a line. You will need ● a compass ● a straightedge P B P A Stage 1 Stage 2 Draw a line and a point labeled P not on the line, as shown above. Describe the construction steps you take at Stage 2. |
How is PA related to PB? What does this answer tell you about where point P lies? Hint: See the Converse of the Perpendicular Bisector Conjecture. Construct the perpendicular bisector of AB. Label the midpoint M. Step 1 Step 2 Step 3 Step 4 152 CHAPTER 3 Using Tools of Geometry Step 5 You have now constructed a perpendicular through a point not on the line. This is useful for finding the distance to a line. Label three randomly placed points on AB as Q, R, and S. Measure PQ, PR, PS, and PM. Which distance is shortest? Compare results with others in your group. P You are now ready to state your observations by completing the conjecture. B S M Q R A Shortest Distance Conjecture The shortest distance from a point to a line is measured along the? from the point to the line. C-7 Let’s take another look. How could you use patty paper to do this construction? Investigation 2 Patty-Paper Perpendiculars In Investigation 1, you constructed a perpendicular from a point to a line. Now let’s do the same construction using patty paper. On a piece of patty paper, perform the steps below. You will need ● patty paper ● a straightedge B P A B A P B M A Step 1 Step 2 Step 3 Step 1 Step 2 Step 3 Draw and label AB and a point P not on AB. Fold the line onto itself, and slide the layers of paper so that point P appears to be on the crease. Is the crease perpendicular to the line? Check it with the corner of a piece of patty paper. Label the point of intersection M. Are AMP and BMP congruent? Supplementary? Why or why not? In Investigation 2, is M the midpoint of AB? Do you think it needs to be? Think about the techniques used in the two investigations. How do the techniques differ? LESSON 3.3 Constructing Perpendiculars to a Line 153 The construction of a perpendicular from a point to a line lets you find the shortest distance from a point to a line. The geometry definition of distance from a point to a line is based on this construction, and it reads, “The distance from a point to a line is the length of the perpendicular segment from the point to the line.” You can also use this construction to find the altitude of a triangle. An altitude of a triangle |
is a perpendicular segment from a vertex to the opposite side or to a line containing the opposite side. Altitude Altitude Altitudes An altitude can be inside the triangle. An altitude can be outside the triangle. An altitude can be one of the sides of the triangle. The length of the altitude is the height of the triangle. A triangle has three different altitudes, so it has three different heights. EXERCISES Construction Use your compass and straightedge and the definition of distance to do Exercises 1–5. You will need Construction tools for Exercises 1–12 1. Draw an obtuse angle BIG. Place a point P inside the angle. Now construct perpendiculars from the point to both sides of the angle. Which side is closer to point P? 2. Draw an acute triangle. Label it ABC. Construct altitude CD with point D on AB. (We didn’t forget about point D. It’s at the foot of the perpendicular. Your job is to locate it.) 3. Draw obtuse triangle OBT with obtuse angle O. Construct altitude BU. In an obtuse triangle, an altitude can fall outside the triangle. To construct an altitude from point B of your triangle, extend side OT. In an obtuse triangle, how many altitudes fall outside the triangle and how many fall inside the triangle? In this futuristic painting, American artist Ralston Crawford (1906–1978) has constructed a set of converging lines and vertical lines to produce an illusion of distance. 4. How can you construct a perpendicular to a line through a point that is on the line? Draw a line. Mark a point on your line. Now experiment. Devise a method to construct a perpendicular to your line at the point. 5. Draw a line. Mark two points on the line and label them Q and R. Now construct a square SQRE with QR as a side. 154 CHAPTER 3 Using Tools of Geometry Construction For Exercises 6–9, use patty paper. (Attach your patty-paper work to your problems.) 6. Draw a line across your patty paper with a straightedge. Place a point P not on the line, and fold the perpendicular to the line through the point P. How would you fold to construct a perpendicular through a point on a line? Place a point Q on the line. Fold a perpendicular to the line through point Q. What do you notice about the two folds? 7. Draw |
a very large acute triangle on your patty paper. Place a point inside the triangle. Now construct perpendiculars from the point to all three sides of the triangle by folding. Mark your figure. How can you use your construction to decide which side of the triangle your point is closest to? 8. Construct an isosceles right triangle. Label its vertices A, B, and C, with point C the right angle. Fold to construct the altitude CD. What do you notice about this line? 9. Draw obtuse triangle OBT with angle O obtuse. Fold to construct the altitude BU. (Don’t forget, you must extend the side OT.) Construction For Exercises 10–12, you may use either patty paper or a compass and a straightedge. 10. Construct a square ABLE given AL as a diagonal. 11. Construct a rectangle whose width is half its length. 12. Construct the complement of A. A L A Review 13. Copy and complete the table. Make a conjecture for the value of the nth term and for the value of the 35th term. Rectangular pattern with triangles Rectangle Number of shaded triangles … 35 … 1 4 3 5 6 7 8 10 14. Sketch the solid of revolution formed when the two-dimensional figure at right is revolved about the line. LESSON 3.3 Constructing Perpendiculars to a Line 155 For Exercises 15–18, label the vertices with the appropriate letters. When you sketch or draw, use the special marks that indicate right angles, parallel segments, and congruent segments and angles. 15. Sketch obtuse triangle FIT with mI 90° and median IY. 16. Sketch AB CD and EF CD. 17. Use your protractor to draw a regular pentagon. Draw all the diagonals. Use your compass to construct a regular hexagon. Draw three diagonals connecting alternating vertices. Do the same for the other three vertices. 18. Draw a triangle with a 6 cm side and an 8 cm side and the angle between them measuring 40°. Draw a second triangle with a 6 cm side and an 8 cm side and exactly one 40° angle that is not between the two given sides. Are the two triangles congruent? IMPROVING YOUR VISUAL THINKING SKILLS Constructing an Islamic Design This Islamic design is based on two intersecting squares that form an 8-pointed star |
. Most Islamic designs of this kind can be constructed using only a compass and a straightedge. Try it. Use your compass and straightedge to re-create this design or to create a design of your own based on an 8-pointed star. Here are two diagrams to get you started. 156 CHAPTER 3 Using Tools of Geometry L E S S O N 3.4 Challenges make you discover things about yourself that you never really knew. CICELY TYSON Constructing Angle Bisectors In Chapter 1, you learned that an angle bisector divides an angle into two congruent angles. While the definition states that the bisector of an angle is a ray, you may also refer to a segment as an angle bisector if the segment lies on the ray and passes through the vertex. For example, in this triangle the two angle bisectors are both segments. In Investigations 1 and 2, you will learn to construct the bisector of an angle. You will need ● patty paper Investigation 1 Angle Bisecting by Folding Each person should draw his or her own acute angle for this investigation. P P Step 1 Step 2 Step 3 Step 4 Step 5 Q R Q P R Q R Step 1 Step 2 Step 3 On patty paper, draw a large-scale angle. Label it PQR. Fold your patty paper so that QP and QR coincide. Crease the paper along the fold. Unfold your patty paper. Draw a ray with endpoint Q along the crease. Does the ray bisect PQR? How can you tell? Repeat Steps 1–3 with an obtuse angle. Do you use different methods for finding the bisectors of different kinds of angles? Place a point on your angle bisector. Label it A. Compare the distances from A to each of the two sides. Remember that “distance” means shortest distance! Try it with other points on the angle bisector. Compare your results with those of others. Copy and complete the conjecture. Angle Bisector Conjecture If a point is on the bisector of an angle, then it is? from the sides of the angle. C-8 LESSON 3.4 Constructing Angle Bisectors 157 You’ve found the bisector of an angle by folding patty paper. Now let’s see how you can construct the angle bisector with a compass and a straightedge. Investigation 2 Angle Bisecting with Compass In this investigation, you will |
find a method for bisecting an angle using a compass and straightedge. Each person in your group should investigate a different angle. You will need ● a compass ● a straightedge Step 1 Step 2 Step 3 Draw an angle. Find a method for constructing the bisector of the angle. Experiment! Hint: Start by drawing an arc centered at the vertex. Once you think you have constructed the angle bisector, fold your paper to see if the ray you constructed is actually the bisector. Share your method with other students in your group. Agree on a best method. Step 4 Write a summary of what you did in this investigation. In earlier lessons, you learned to construct a 90° angle. Now you know how to bisect an angle. What angles can you construct by combining these two skills? EXERCISES Construction For Exercises 1–5, match each geometric construction with its diagram. 1. Construction of an angle bisector 2. Construction of a median You will need Construction tools for Exercises 1–12 Geometry software for Exercise 17 3. Construction of a midsegment 4. Construction of a 5. Construction of an altitude perpendicular bisector A. D. B. E. C. F. 158 CHAPTER 3 Using Tools of Geometry Construction For Exercises 6–12, construct a figure with the given specifications. 6. Given: z Construct: An isosceles right triangle with z as the length of each of the two congruent sides 7. Given: A R P A R Construct: RAP with median PM and angle bisector RB P 8. Given: M M M E S Construct: MSE with OU, where O is the midpoint of MS and U is the midpoint of SE 9. Construct an angle with each given measure and label it. Remember, you may use only your compass and straightedge. No protractor! a. 90° b. 45° c. 135° 10. Draw a large acute triangle. Bisect one vertex with a compass and a straightedge. Construct an altitude from the second vertex and a median from the third vertex. 11. Repeat Exercise 10 with patty paper. Which set of construction tools do you prefer? Why? 12. Use your straightedge to draw a linear pair of angles. Use your compass to bisect each angle of the linear pair. What do you notice about the two angle bisectors? Can you make a conjecture? Can you think of a way to explain why |
it is true? 13. In this lesson you discovered the Angle Bisector Conjecture. Write the converse of the Angle Bisector Conjecture. Do you think it’s true? Why or why not? Notice how this mosaic floor at Church of Pomposa in Italy (ca. 850 C.E.) uses many duplicated shapes. What constructions do you see in the square pattern? Are all the triangles in the isosceles triangle pattern identical? How can you tell? LESSON 3.4 Constructing Angle Bisectors 159 Review Sketch, draw, or construct each figure in Exercises 14–17. Label the vertices with the appropriate letters. 14. Draw a regular octagon. What traffic 15. Construct regular octagon sign comes to mind? ALTOSIGN. 16. Draw a triangle with a 40° angle, a 60° angle, and a side between the given angles measuring 8 cm. Draw a second triangle with a 40° angle and a 60° angle but with a side measuring 8 cm opposite the 60° angle. Are the triangles congruent? 17. Technology Use geometry software to construct AB and CD, with point C on AB and point D not on AB. Construct the perpendicular bisector of CD. a. Trace this perpendicular bisector as you drag point C along AB. Describe the shape formed by this locus of lines. b. Erase the tracings from part a. Now trace the midpoint of CD as you drag C. Describe the locus of points. IMPROVING YOUR VISUAL THINKING SKILLS Coin Swap III Arrange four dimes and four pennies in a row of nine squares, as shown. Switch the position of the four dimes and four pennies in exactly 24 moves. A coin can slide into an empty square next to it or can jump over one coin into an empty space. Record your solution by listing, in order, which type of coin is moved. For example, your list might begin PDPDPPDD.... 160 CHAPTER 3 Using Tools of Geometry Constructing Parallel Lines Parallel lines are lines that lie in the same plane and do not intersect. L E S S O N 3.5 When you stop to think, don’t forget to start up again. ANONYMOUS The lines in the first pair shown above intersect. They are clearly not parallel. The lines in the second pair do not meet as drawn. However |
, if they were extended, they would intersect. Therefore, they are not parallel. The lines in the third pair appear to be parallel, but if you extend them far enough in both directions, can you be sure they won’t meet? There are many ways to be sure that the lines are parallel. You will need ● patty paper Investigation Constructing Parallel Lines by Folding How would you check whether two lines are parallel? One way is to draw a transversal and compare corresponding angles. You can also use this idea to construct a pair of parallel lines. Step 1 Step 2 Step 1 Step 2 Draw a line and a point on patty paper as shown. Fold the paper to construct a perpendicular so that the crease runs through the point as shown. Describe the four newly formed angles. Step 3 Step 4 Step 3 Step 4 Through the point, make another fold that is perpendicular to the first crease. Match the pairs of corresponding angles created by the folds. Are they all congruent? Why? What conclusion can you make about the lines? LESSON 3.5 Constructing Parallel Lines 161 There are many ways to construct parallel lines. You can construct parallel lines much more quickly with patty paper than with compass and straightedge. You can also use properties you discovered in the Parallel Lines Conjecture to construct parallel lines by duplicating corresponding angles, alternate interior angles, or alternate exterior angles. Or you can construct two perpendiculars to the same line. In the exercises you will practice all of these methods. EXERCISES Construction In Exercises 1–9, use the specified construction tools to do each construction. If no tools are specified, you may choose either patty paper or compass and straightedge. You will need Construction tools for Exercises 1–9 1. Use compass and straightedge. Draw a line and a point not on the line. Construct a second line through the point that is parallel to the first line, by duplicating alternate interior angles. 2. Use compass and straightedge. Draw a line and a point not on the line. Construct a second line through the point that is parallel to the first line, by duplicating corresponding angles. 3. Construct a square with perimeter z. z 4. Construct a rhombus with x as the length of each side and A as one of the acute angles. x A 5. Construct trapezoid TRAP with TR and AP as the two parallel sides and with AP as the distance between them. (There |
are many solutions!) T A R P 6. Using patty paper and straightedge, or a compass and straightedge, construct parallelogram GRAM with RG and RA as two consecutive sides and ML as the distance between RG and AM. (How many solutions can you find?) G R M R A L 162 CHAPTER 3 Using Tools of Geometry 7. Mini-Investigation Construct a large scalene acute triangle and label it SUM. Through vertex M construct a line parallel to side SU as shown in the diagram. Use your protractor or a piece of patty paper to compare 1 and 2 with the other two angles of the triangle (S and U ). Notice anything special? Can you explain why? 8. Mini-Investigation Construct a large scalene acute triangle and label it PAR. Place point E anywhere on side PR, and construct a line EL parallel to side PA as shown in the diagram. Use your ruler to measure the lengths of the four segments AL, LR, RE, and EP, and compare ratios E. Notice anything special? L and R R EP A L (You may also do this problem using geometry software.) 9. Mini-Investigation Draw a pair of parallel lines by tracing along both edges of your ruler. Draw a transversal. Use your compass to bisect each angle of a pair of alternate interior angles. What shape is formed? Can you explain why Review 10. There are three fire stations in the small county of Dry Lake. County planners need to divide the county into three zones so that fire alarms alert the closest station. Trace the county and the three fire stations onto patty paper, and locate the boundaries of the three zones. Explain how these boundaries solve the problem. 11. Copy the diagram below. Use your conjectures to calculate the measure of each lettered angle. j h k g 108 118° q p r LESSON 3.5 Constructing Parallel Lines 163 Sketch or draw each figure in Exercises 12–14. Label the vertices with the appropriate letters. Use the special marks that indicate right angles, parallel segments, and congruent segments and angles. 12. Sketch trapezoid ZOID with ZO ID, point T the midpoint of OI, and R the midpoint of ZD. Sketch segment TR. 13. Draw rhombus ROMB with mR 60° and diagonal OB. 14. Draw rectangle RECK with diagonals RC and EK both 8 cm long and intersecting at point W |
. IMPROVING YOUR VISUAL THINKING SKILLS Visual Analogies Which of the designs at right complete the statements at left? Explain. 1. is to as is to? 2. is to as is to? 3. is to as is to? A. C. A. C. A. C. B. D. B. D. B. D. 164 CHAPTER 3 Using Tools of Geometry ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 3 ● USING YO USING YOUR ALGEBRA SKILLS 3 USING YOUR ALGEBRA SKILLS 1 Slopes of Parallel and Perpendicular Lines If two lines are parallel, how do their slopes compare? If two lines are perpendicular, how do their slopes compare? In this lesson you will review properties of the slopes of parallel and perpendicular lines. 3 4 If the slopes of two or more distinct lines are equal, are the lines parallel? To find out, try drawing on graph paper two lines that have the same slope triangle. Yes, the lines are parallel. In fact, in coordinate geometry, this is the definition of parallel lines. The converse of this is true as well: If two lines are parallel, their slopes must be equal. 4 3 Parallel Slope Property In a coordinate plane, two distinct lines are parallel if and only if their slopes are equal. If two lines are perpendicular, their slope triangles have a different relationship. Study the slopes of the two perpendicular lines at right. y x Slope 3_ 4 Slope 4_ 3 3 4 4 3 Perpendicular Slope Property In a coordinate plane, two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. Can you explain why the slopes of perpendicular lines would have opposite signs? Can you explain why they would be reciprocals? Why do the lines need to be nonvertical? USING YOUR ALGEBRA SKILLS 3 Slopes of Parallel and Perpendicular Lines 165 ALGEBRA SKILLS 3 ● USING YOUR ALGEBRA SKILLS 3 ● USING YOUR ALGEBRA SKILLS 3 ● USING YO EXAMPLE A Consider A(15, 6), B(6, 8), C(4, 2), and D(4, 10). Are AB and CD parallel, perpendicular, or neither? Solution Calculate |
the slope of each line. ) 6 ( 2 slope of AB 3 ( ) 5 1, are negative reciprocals of each other, so AB CD. and 3 The slopes, 2 2 3 ( 2) 3 slope of CD 10 4 4 2 8 6 EXAMPLE B Given points E(3, 0), F(5, 4), and Q(4, 2), find the coordinates of a point P such that PQ is parallel to EF. y Solution We know that if PQ EF, then the slope of PQ equals the slope of EF. First find the slope of EF. 4 1 0 4 slope of EF 3 2 8 ( ) 5 E (–3, 0) Q (4, 2) F (5, –4) x There are many possible ordered pairs (x, y) for P. Use (x, y) as the coordinates of P, and the given coordinates of Q, in the slope formula to get 2 y 1 4 2 x 4 x 2 Now you can treat the denominators and numerators as separate equations Language Coordinate geometry is sometimes called “analytic geometry.” This term implies that you can use algebra to further analyze what you see. For example, consider AB and CD. They look parallel, but looks can be deceiving. Only by calculating the slopes will you see that the lines are not truly parallel. Thus one possibility is P(2, 3). How could you find another ordered pair for P? Here’s a hint: How many different ways can you express 1? 2 y (10, 6) B D (10, 3) x (–15, –6) A C (–2, –3) 166 CHAPTER 3 Using Tools of Geometry ALGEBRA SKILLS 3 ● USING YOUR ALGEBRA SKILLS 3 ● USING YOUR ALGEBRA SKILLS 3 ● USING YO EXERCISES For Exercises 1–4, determine whether each pair of lines through the points given below is parallel, perpendicular, or neither. B(3, 4) A(1, 2) 1. AB and BC C(5, 2) D(8, 3) 2. AB and CD E(3, 8) F(6, 5) 3. AB and DE 4. CD and EF 5. Given A(0, 3), B(5, 3), and Q(3, 1), find two possible locations for a point |
P such that PQ is parallel to AB. 6. Given C(2, 1), D(5, 4), and Q(4, 2), find two possible locations for a point P such that PQ is perpendicular to CD. For Exercises 7–9, find the slope of each side, and then determine whether each figure is a trapezoid, a parallelogram, a rectangle, or just an ordinary quadrilateral. Explain how you know. 7. y 10 E T M 8. T y 3 9. E I x 10 10. Quadrilateral HAND has vertices H(5, 1), A(7, 1), N(6, 7), and D(6, 5). a. Is quadrilateral HAND a parallelogram? A rectangle? Neither? Explain how you know. b. Find the midpoint of each diagonal. What can you conjecture? 11. Quadrilateral OVER has vertices O(4, 2), V(1, 1), E(0, 6), and R(5, 7). a. Are the diagonals perpendicular? Explain how you know. b. Find the midpoint of each diagonal. What can you conjecture? c. What type of quadrilateral does OVER appear to be? Explain how you know. 12. Consider the points A(5, 2), B(1, 1), C(1, 0), and D(3, 2). a. Find the slopes of AB and CD. b. Despite their slopes, AB and CD are not parallel. Why not? c. What word in the Parallel Slope Property addresses the problem in 12b? 13. Given A(3, 2), B(1, 5), and C(7, 3), find point D such that quadrilateral ABCD is a rectangle. USING YOUR ALGEBRA SKILLS 3 Slopes of Parallel and Perpendicular Lines 167 L E S S O N 3.6 People who are only good with hammers see every problem as a nail. ABRAHAM MASLOW Construction Problems Once you know the basic constructions, you can create more complex geometric figures. You know how to duplicate segments and angles with a compass and straightedge. If given a triangle, you can use these two constructions to duplicate the triangle by copying each segment and angle. Can you construct a triangle if you are given the parts separately? Would you need all six parts—three segments and three angles |
— to construct a triangle? Let’s first consider a case in which only three segments are given. EXAMPLE A Construct ABC using the three segments AB, BC, and CA shown below. How many different-size triangles can be drawn? A B C B C A Solution You can begin by copying one segment, for example AC. Then adjust your compass to match the length of another segment. Using this length as a radius, draw a circle centered at one endpoint of the first segment. Now use the third segment length as the radius for another circle, this one centered at the other endpoint. The third vertex of the triangle is where the circles intersect. C B A In the construction above, the segment lengths determine the sizes of the circles and where they intersect. Once the triangle “closes” at the intersection of the arcs, the angles are determined, too. So the lengths of the segments affect the size of the angles. There is only one size of triangle that can be drawn with the segments given, so the segments determine the triangle. Does having three angles also determine a triangle? EXAMPLE B Construct ABC with patty paper by copying the three angles A, B, and C shown below. How many different size triangles can be drawn? A B Solution In this patty-paper construction the angles do not determine the segment length. You can locate the endpoint of a segment anywhere along an angle’s side without affecting the angle measures. Infinitely many different triangles can be drawn with the angles given. Here are just a few examples. C C A B 168 CHAPTER 3 Using Tools of Geometry C A B B A C B C A The angles do not determine a particular triangle. Since a triangle has a total of six parts, there are several combinations of segments and angles that may or may not determine a triangle. Having one or two parts given is not enough to determine a triangle. Is having three parts enough? That answer depends on the combination. In the exercises you will construct triangles and quadrilaterals with various combinations of parts given. EXERCISES Construction In Exercises 1–10, first sketch and label the figure you are going to construct. Second, construct the figure, using either a compass and straightedge, or patty paper and straightedge. Third, describe the steps in your construction in a few sentences. You will need Construction tools for Exercises 1–10 Geometry software for Exercise 11 1. Given: M A M Construct: MAS 2 |
. Given: O Construct: DOT 3. Given: Y Construct: IGY LESSON 3.6 Construction Problems 169 4. Given the triangle shown at right, construct another triangle with angles congruent to the given angles but with sides not congruent to the given sides. Is there more than one triangle with the same three angles? 5. The two segments and the angle below do not determine a triangle. Given: A B C B A Construct: Two different (noncongruent) triangles named ABC that have the three given parts 6. Given: x y Construct: Isosceles triangle CAT with perimeter y and length of the base equal to x 7. Construct a kite. 8. Construct a quadrilateral with two pairs of opposite sides of equal length. 9. Construct a quadrilateral with exactly three sides of equal length. 10. Construct a quadrilateral with all four sides of equal length. 11. Technology Using geometry software, draw a large scalene obtuse triangle ABC with B the obtuse angle. Construct the angle bisector BR, the median BM, and the altitude BS. What is the order of the points on AC? Drag B. Is the order of points always the same? Write a conjecture. B A R C Review 12. Draw each figure and decide how many reflectional and rotational symmetries it has. Copy and complete the table at right. 170 CHAPTER 3 Using Tools of Geometry Reflectional symmetries Rotational symmetries Figure Trapezoid Kite Parallelogram Rhombus Rectangle 13. Draw the new position of TEA if it is reflected over the dotted line. Label the coordinates of the vertices. 14. Sketch the three-dimensional figure formed by folding the net below into a solid. y 5 E –5 –5 T x A 15. If a polygon has 500 diagonals from each vertex, how many sides does it have? IMPROVING YOUR REASONING SKILLS Spelling Card Trick This card trick uses one complete suit (hearts, clubs, spades, or diamonds) from a regular deck of playing cards. How must you arrange the cards so that you can successfully complete the trick? Here is what your audience should see and hear as you perform. 1. As you take the top card and place it at the bottom of the pile, say “A.” 2. Then take the second card, place it at the bottom of the |
pile, and say “C.” 3. Take the third card, place it at the bottom, and say “E.” 4. You’ve just spelled ace. Now take the fourth card and turn it faceup on the table. The card should be an ace. 5. Continue in this fashion, saying “T,” “W,” and “O” for the next three cards. Then turn the next card faceup. It should be a 2. 6. Continue spelling three, four,..., jack, queen, king. Each time you spell a card, the next card turned faceup should be that card. LESSON 3.6 Construction Problems 171 L E S S O N 1.0 Perspective Drawing You know from experience that when you look down a long straight road, the parallel edges and the center line seem to meet at a point on the horizon. To show this effect in a drawing, artists use perspective, the technique of portraying solid objects and spatial relationships on a flat surface. Renaissance artists and architects in the fifteenth century developed perspective, turning to geometry to make art appear true-to-life. In a perspective drawing, receding parallel lines (lines that run directly away from the viewer) converge at a vanishing point on the horizon line. Locate the horizon line, the vanishing point, and converging lines in the perspective study below by Jan Vredeman de Vries. (Below) Perspective study by Dutch artist Jan Vredeman de Vries (1527–1604) 172 CHAPTER 3 Using Tools of Geometry You will need ● a ruler Activity Boxes in Space In this activity you’ll learn to draw a box in perspective. Perspective drawing is based on the relationships between many parallel and perpendicular lines. The lines that recede to the horizon make you visually think of parallel lines even though they actually intersect at a vanishing point. First, you’ll draw a rectangular solid, or box, in one-point perspective. Look at the diagrams below for each step. V h V h Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 1 Step 2 Step 3 Step 4 Draw a horizon line h and a vanishing point V. Draw the front face of the box with its horizontal edges parallel to h. Connect the corners of the box face to V with dashed lines. Draw the upper rear box edge parallel to h. Its endpoints determine the vertical edges of the back |
face. Draw the hidden back vertical and horizontal edges with dashed lines. Erase unnecessary lines and dashed segments. Repeat Steps 1–4 several more times, each time placing the first box face in a different position with respect to h and V—above, below, or overlapping h; to the left or right of V or centered on V. Share your drawings in your group. Tell which faces of the box recede and which are parallel to the imaginary window or picture plane that you see through. What is the shape of a receding box face? Think of each drawing as a scene. Where do you seem to be standing to view each box? That is, how is the viewing position affected by placing V to the left or right of the box? Above or below the box? EXPLORATION Perspective Drawing 173 You can also use perspective to play visual tricks. The Italian architect Francesco Borromini (1599–1667) designed and built a very clever colonnade in the Palazzo Spada. The colonnade is only 12 meters long, but he made it look much longer by designing the sides to get closer and closer to each other and the height of the columns to gradually shrink. If the front surface of a box is not parallel to the picture plane, then you need two vanishing points to show the two front faces receding from view. This is called two-point perspective. Let’s look at a rectangular solid with one edge viewed straight on. Step 7 Draw a horizon line h and select two vanishing points on it, V1 and V2. Draw a vertical segment for the nearest box edge. V1 V2 h Step 8 Connect each endpoint of the box edge to V1 and V2 with dashed lines. V1 V2 h Step 9 Draw two vertical segments within the dashed lines as shown. Connect their endpoints to the endpoints of the front edge along the dashed lines. Now you have determined the position of the hidden back edges that recede from view. V1 V2 h 174 CHAPTER 3 Using Tools of Geometry Step 10 Draw the remaining edges along vanishing lines, using dashed lines for hidden edges. Erase unnecessary dashed segments. V1 V2 h Step 11 Repeat Steps 7–10 several times, each time placing the nearest box edge in a different position with respect to h, V1, and V2, and varying the distance between V1, and V2. You can also experiment with different-shaped boxes. Step 12 Share your drawings |
in your group. Are any faces of the box parallel to the picture plane? Does each box face have a pair of parallel sides? Explain how the viewing position is affected by the distance between V1 and V2 relative to the size of the box. Must the box be between V1 and V2? Perspective helps in designing the lettering painted on streets. From above, letters appear tall, but from a low angle they appear normal. Tilt the page up to your face. How do the letters look? EXPLORATION Perspective Drawing 175 L E S S O N 3.7 Nothing in life is to be feared, it is only to be understood. MARIE CURIE Constructing Points of Concurrency You now can perform a number of constructions in triangles, including angle bisectors, perpendicular bisectors of the sides, medians, and altitudes. In this lesson and the next lesson you will discover special properties of these lines and segments. When three or more lines have a point in common, they are concurrent. Segments, rays, and even planes are concurrent if they intersect in a single point. Point of concurrency Not concurrent Concurrent The point of intersection is the point of concurrency. You will need ● patty paper ● a compass ● a straightedge Investigation 1 Concurrence In this investigation you will discover that some special lines in a triangle have points of concurrency. You should investigate each set of lines on an acute triangle, an obtuse triangle, and a right triangle to be sure your conjecture applies to all triangles. Step 1 Draw a large acute triangle on one patty paper and an obtuse triangle on another. If you’re using a compass and a straightedge, draw your triangles on the top and bottom halves of a piece of paper. Step 2 Construct the three angle bisectors for each triangle. Are they concurrent? Compare your results with the results of others. State your observations as a conjecture. Angle Bisector Concurrency Conjecture The three angle bisectors of a triangle?. C-9 Step 3 Construct the perpendicular bisector for each side of the triangle and complete the conjecture. 176 CHAPTER 3 Using Tools of Geometry Perpendicular Bisector Concurrency Conjecture The three perpendicular bisectors of a triangle?. C-10 Step 4 Construct the lines containing the altitudes of your triangle and complete the conjecture. Altitude Concurrency Conjecture The three altitudes (or the lines containing the altitudes) of |
a triangle?. C-11 Step 5 For what kind of triangle will the points of concurrency be the same point? The point of concurrency for the three angle bisectors is the incenter. The point of concurrency for the perpendicular bisectors is the circumcenter. The point of concurrency for the three altitudes is called the orthocenter. You will investigate a triangle’s medians in the next lesson. Rudolf Bauer (1889–1953) titled this painting Rounds and Triangles. Investigation 2 Incenter and Circumcenter In this investigation you will discover special properties of the incenter and the circumcenter. Measure and compare the distances from the circumcenter to each of the three vertices. Are they the same? Compare the distances from the circumcenter to each of the three sides. Are they the same? State your observations as your next conjecture. You will need ● construction tools Step 1 Circumcenter Conjecture The circumcenter of a triangle?. C-12 LESSON 3.7 Constructing Points of Concurrency 177 Step 2 Measure and compare the distances from the incenter to each of the three sides. Are they the same? State your observations as your next conjecture. Incenter Conjecture The incenter of a triangle?. C-13 You just discovered a very useful property of the circumcenter and a very useful property of the incenter. You will see some applications of these properties in the exercises. With earlier conjectures and logical reasoning, you can explain why your conjectures are true. Let’s look at a paragraph proof of the Circumcenter Conjecture. Paragraph Proof of the Circumcenter Conjecture If the Perpendicular Bisector Conjecture is true, then you know that each point on a perpendicular bisector of a segment is equidistant from the endpoints of the segment. So, in the figure at left, since point P lies on the perpendicular bisector of AL, it is equidistant from vertices A and L. Point P also lies on the perpendicular bisector of LY, so it is also equidistant from vertices L and Y. Therefore P is equidistant from all three vertices. In other words, the circumcenter of a triangle is on all three perpendicular bisectors, so it follows logically that it is equidistant from all three vertices of the triangle. Your investigation led you to believe the conjecture is true, and the logical argument |
helps to understand why it is true. Y L 1 A P 2 You can use your compass to construct a circle that passes through the three vertices of the triangle with the circumcenter as the center of the circle. So, you can also think of the circumcenter as the center of a circle that passes through the three vertices of a triangle. You can make a similar logical argument for the Incenter Conjecture. Paragraph Proof of the Incenter Conjecture R Y Da c b L A H If the Angle Bisector Conjecture is true, then you know that each point on an angle bisector is equidistant from the sides of the angle. So, since point D lies on the angle bisector of RHA, it is equidistant from RH and HA. Point D also lies on RL, the angle bisector of HRA, so it is also equidistant from RH and RA. So point D is equidistant from all three sides. In other words, the incenter of a triangle is on all three angle bisectors. It follows logically that the incenter is equidistant from all three sides. Again your investigation may have convinced you that this was true, but the logical argument explains why it is true. 178 CHAPTER 3 Using Tools of Geometry You can use your compass to construct a circle that is tangent to the three sides with the incenter as the center of a circle. To construct the circle you need the radius, the shortest distance from the incenter to each side. To get the radius, you construct the perpendicular from the incenter to one of the sides. The incenter is the center of a circle that touches each side of the triangle. Here are a few vocabulary terms that help describe these geometric situations. A circle is circumscribed about a polygon if and only if it passes through each vertex of the polygon. (The polygon is inscribed in the circle.) A circle is inscribed in a polygon if and only if it touches each side of the polygon at exactly one point. (The polygon is circumscribed about the circle.) Circumscribed circle (inscribed triangle) Inscribed circle (circumscribed triangle) This geometric art by geometry student Ryan Garvin shows the construction of the incenter, its perpendicular distance to one side of the triangle, and the inscribed circle. EXERCISES For Exercises 1–4, make a sketch and explain how to find the answer. 1. The first- |
aid center of Mt. Thermopolis State Park needs to be at a point that is equidistant from three bike paths that intersect to form a triangle. Locate this point so that in an emergency medical personnel will be able to get to any one of the paths by the shortest route possible. Which point of concurrency is it? You will need Construction tools for Exercises 6, 7, 12–15, and 17 Geometry software for Exercises 10, 11, and 18 2. Rosita wants to install a circular sink in her new triangular countertop. She wants to choose the largest sink that will fit. Which point of concurrency must she locate? Explain. 3. Julian Chive wishes to center a butcher-block table at a location equidistant from the refrigerator, stove, and sink. Which point of concurrency does Julian need to locate? LESSON 3.7 Constructing Points of Concurrency 179 Art The designer of stained glass arranges pieces of painted glass to form the elaborate mosaics that you might see in Gothic cathedrals or on Tiffany lampshades. He first organizes the glass pieces by shape and color according to the design. He mounts these pieces into a metal framework that will hold the design. With precision, the designer cuts every glass piece so that it fits against the next one with a strip of cast lead. The result is a pleasing combination of colors and shapes that form a luminous design when viewed against light. 4. A stained-glass artist wishes to circumscribe a circle about a triangle in her latest abstract design. Which point of concurrency does she need to locate? 5. One event at this year’s Battle of the Classes will be a pie-eating contest between the sophomores, juniors, and seniors. Five members of each class will be positioned on the football field at the points indicated below. At the whistle, one student from each class will run to the pie table, eat exactly one pie, and run back to his or her group. The next student will then repeat the process. The first class to eat five pies and return to home base will be the winner of the pie-eating contest. Where should the pie table be located so that it will be a fair contest? Describe how the contest planners should find that point. S R A G U O C Sophomores C Seniors Juniors 20 50 20 10 50 30 30 40 40 10 40 40 30 30 20 10 20 10 C O U G A |
R S 6. Construction Draw a large triangle. Construct a circle inscribed in the triangle. 7. Construction Draw a triangle. Construct a circle circumscribed about the triangle. 8. Is the inscribed circle the greatest circle to fit within a given triangle? Explain. If you think not, give a counterexample. 9. Does the circumscribed circle create the smallest circular region that contains a given triangle? Explain. If you think not, give a counterexample. 10. Technology Notice that the circumcenter is in the interior of acute triangles and on the exterior of obtuse triangles. Use geometry software to construct a right triangle. Where is the circumcenter of a right triangle? (This problem can also be done by drawing several right triangles on graph paper.) 11. Technology Notice that the orthocenter is in the interior of acute triangles and on the exterior of obtuse triangles. Use geometry software to construct a right triangle. Where is the orthocenter of a right triangle? (This problem can also be done by drawing several right triangles on graph paper.) 180 CHAPTER 3 Using Tools of Geometry Review Construction Use the segments and angle at right to construct each figure in Exercises 12–15. 12. Mini-Investigation Construct MAT. Construct H the midpoint of MT and S the midpoint of AT. Construct the midsegment HS. Compare the lengths of HS and MA. Notice anything special? T M A T T 13. Mini-Investigation An isosceles trapezoid is a trapezoid with the nonparallel sides congruent. Construct isosceles trapezoid MOAT with MT OA and AT MO. Use patty paper to compare T and M. Notice anything special? 14. Mini-Investigation Construct a circle with diameter MT. Construct chord TA. Construct chord MA to form MTA. What is the measure of A? Notice anything special? 15. Mini-Investigation Construct a rhombus with TA as the length of a side and T as one of the acute angles. Construct the two diagonals. Notice anything special? 16. Sketch the locus of points on the coordinate plane in which the sum of the x-coordinate and the y-coordinate is 9. 17. Construction Bisect the missing angle of this triangle. How can you do it without re-creating the third angle? 18. Technology Is it possible for the midpoints of the three altitudes of a triangle to be collinear |
? Investigate by using geometry software. Write a paragraph describing your findings. 19. Sketch the section formed when the plane slices the cube as shown. For Exercises 20–24, match each geometric construction with one of the figures below. 20. Construction of a perpendicular bisector 21. Construction of an angle bisector 22. Construction of a perpendicular through 23. Construction of a line parallel to a given line a point on a line through a given point not on the line 24. Construction of an equilateral triangle A. D. B. E. C. LESSON 3.7 Constructing Points of Concurrency 181 IMPROVING YOUR VISUAL THINKING SKILLS The Puzzle Lock This mysterious pattern is a lock that must be solved like a puzzle. Here are the rules: You must make eight moves in the proper sequence. To make each move (except the last), you place a gold coin onto an empty circle, then slide it along a diagonal to another empty circle. You must place the first coin onto circle 1, then slide it to either circle 4 or circle 6. You must place the last coin onto circle 5. You do not slide the last coin. Solve the puzzle. Copy and complete the table to show your solution. Coin Movements Placed on Slid to → → → → → → → 1 5 Coin First Second Third Fourth Fifth Sixth Seventh Eighth 7 6 8 5 1 4 2 3 182 CHAPTER 3 Using Tools of Geometry The Centroid In the previous lesson you discovered that the three angle bisectors are concurrent, the three perpendicular bisectors of the sides are concurrent, and the three altitudes in a triangle are concurrent. You also discovered the properties of the incenter and the circumcenter. In this lesson you will investigate the medians of a triangle. L E S S O N 3.8 The universe may be as great as they say, but it wouldn’t be missed if it didn’t exist. PIET HEIN Three angle bisectors (incenter) Three perpendicular bisectors (circumcenter) Three altitudes (orthocenter)? Three medians? Investigation 1 Are Medians Concurrent? Each person in your group should draw a different triangle for this investigation. Make sure you have at least one acute triangle, one obtuse triangle, and one right triangle in your group. You will need ● patty paper ● a straightedge Step 1 On a sheet of patty paper, draw as large a scal |
ene triangle as possible and label it CNR, as shown at right. Locate the midpoints of the three sides. Construct the medians and complete the conjecture. C Median Concurrency Conjecture The three medians of a triangle?. R N C-14 The point of concurrency of the three medians is the centroid. Step 2 Step 3 Label the three medians CT, NO, and RE. Label the centroid D. Use your compass or another sheet of patty paper to investigate whether there is anything special about the centroid. Is the centroid equidistant from the three vertices? From the three sides? Is the centroid the midpoint of each median? R O T D C E N LESSON 3.8 The Centroid 183 Step 4 Step 5 The centroid divides a median into two segments. Use your patty paper or compass to compare the length of the longer segment to the length of the shorter segment and find the ratio. Find the ratios of the lengths of the segment parts for the other two medians. Do you get the same ratio for each median? R O T D Compare your results with the results of others. State your discovery as a conjecture, and add it to your conjecture list. C E N Centroid Conjecture C-15 The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is? the distance from the centroid to the midpoint of the opposite side. In earlier lessons you discovered that the midpoint of a segment is the balance point or center of gravity. You also saw that when a set of segments is arranged into a triangle, the line through the midpoints of these segments can act as a line of balance for the triangle. Can you then balance a triangle on a median? Let’s take a look.? You will need ● cardboard Investigation 2 Balancing Act Use your patty paper from Investigation 1 for this investigation. Step 1 Step 2 Step 3 Place your patty paper from the previous investigation on a piece of mat board or cardboard. With a sharp pencil tip or compass tip, mark the three vertices, the three midpoints, and the centroid on the board. Draw in the triangle and medians on the cardboard. Cut out the cardboard triangle. Try balancing the triangle on each of the three medians by placing the median on the edge of a ruler. If you are successful, what does that imply about the areas of the two triangles |
formed by one median? Step 4 Is there a single point where you can balance the triangle? 184 CHAPTER 3 Using Tools of Geometry If you have found the balancing point for the triangle, you have found its center of gravity. State your discovery as a conjecture, and add it to your conjecture list. Center of Gravity Conjecture The? of a triangle is the center of gravity of the triangular region. C-16 The triangle balances on each median and the centroid is on each median, so the triangle balances on the centroid. As long as the weight of the cardboard is distributed evenly throughout the triangle, you can balance any triangle at its centroid. For this reason, the centroid is a very useful point of concurrency, especially in physics. You have discovered special properties of three of the four points of concurrency—the incenter, the circumcenter, and the centroid. The incenter is the center of an inscribed circle, the circumcenter is the center of a circumscribed circle, and the centroid is the center of gravity. You can learn more about the orthocenter in the Project Is There More to the Orthocenter? Science In physics, the center of gravity of an object is an imaginary point where the total weight is concentrated. The center of gravity of a tennis ball, for example, would be in the hollow part, not in the actual material of the ball. The idea is useful in designing structures as complicated as bridges or as simple as furniture. Where is the center of gravity of the human body? Incenter Circumcenter Centroid LESSON 3.8 The Centroid 185 EXERCISES 1. Birdy McFly is designing a large triangular hang glider. She needs to locate the center of gravity for her glider. Which point does she need to locate? Birdy wishes to decorate her glider with the largest possible circle within her large triangular hang glider. Which point of concurrency does she need to locate? In Exercises 2–4, use your new conjectures to find each length. You will need Construction tools for Exercises 5 and 6 Geometry software for Exercise 7 2. Point M is the centroid. 3. Point G is the centroid. 4. Point Z is the centroid CM 16 MO 10 TS 21 AM? SM? TM? UM? GI GR GN ER 36 BG? IG? T C H Z A E R CZ 14 TZ 30 RZ AZ RH? TE? |
5. Construction Construct an equilateral triangle, then construct angle bisectors from two vertices, medians from two vertices, and altitudes from two vertices. What can you conclude? 6. Construction On patty paper, draw a large isosceles triangle with an acute vertex angle that measures less than 40°. Copy it onto three other pieces of patty paper. Construct the centroid on one patty paper, the incenter on a second, the circumcenter on a third, and the orthocenter on a fourth. Record the results of all four pieces of patty paper on one piece of patty paper. What do you notice about the four points of concurrency? What is the order of the four points of concurrency from the vertex to the opposite side in an acute isosceles triangle? 7. Technology Use geometry software to construct a large isosceles acute triangle. Construct the four points of concurrency. Hide all constructions except for the points of concurrency. Label them. Drag until it has an obtuse vertex angle. Now what is the order of the four points of concurrency from the vertex angle to the opposite side? When did the order change? Do the four points ever become one? 186 CHAPTER 3 Using Tools of Geometry 8. Where do you think the center of gravity is located on a square? A rectangle? A rhombus? In each case the center of gravity is not that difficult to find, but what about an ordinary quadrilateral? Experiment to discover a method for finding the center of gravity for a quadrilateral by geometric construction. Test your method on a large cardboard quadrilateral. Review 9. Sally Solar is the director of Lunar Planning for Galileo Station on the moon. She has been asked to locate the new food production facility so that it is equidistant from the three main lunar housing developments. Which point of concurrency does she need to locate? 10. Construct circle O. Place an arbitrary point P within the circle. Construct the longest chord passing through P. Construct the shortest chord passing through P. How are they related? 11. A billiard ball is hit so that it travels a distance equal to AB but bounces off the cushion at point C. Copy the figure, and sketch where the ball will rest. A 12. APPLICATION In alkyne molecules all the bonds are single bonds except one triple bond between two carbon atoms. The first three alkynes are modeled below. The dash ( ) between |
letters represents single bonds. The triple dash ( ) between letters represents a triple bond HC H C C C HC H H H Ethyne C2H2 Propyne C3H4 Butyne C4H6 Sketch the alkyne with eight carbons in the chain. What is the general rule for alkynes CnH? hydrogen atoms (H) are in the alkyne?? In other words, if there are n carbon atoms (C), how many C B LESSON 3.8 The Centroid 187 13. When plane figure A is rotated about the line it produces the solid figure B. What is the plane figure that produces the solid figure D? A B? C D 14. Copy the diagram below. Use your Vertical Angles Conjecture and Parallel Lines Conjecture to calculate each lettered angle measure. b c a 128 15. A brother and a sister have inherited a large triangular plot of land. The will states that the property is to be divided along the altitude from the northernmost point of the property. However, the property is covered with quicksand at the northern vertex. The will states that the heir who figures out how to draw the altitude without using the northern vertex point gets to choose his or her parcel first. How can the heirs construct the altitude? Is this a fair way to divide the land? Why or why not? Quicksand N W E S ngelo A n a S e k a L Property Boundary 16. At the college dorm open house, each of the 20 dorm members brings two guests (usually their parents). How many greetings are possible if you do not count dorm members greeting their own guests? IMPROVING YOUR REASONING SKILLS The Dealer’s Dilemma In the game of bridge, the dealer deals 52 cards in a clockwise direction among four players. You are playing a game in which you are the dealer. You deal the cards, starting with the player on your left. However, in the middle of dealing you stop to answer the phone. When you return, no one can remember where the last card was dealt. (And, of course, no cards have been touched.) Without counting the number of cards in anyone’s hand or the number of cards yet to be dealt, how can you rapidly finish dealing, giving each player exactly the same cards she or he would have received if you hadn’t been interrupted? 188 CHAPTER 3 Using Tools of Geometry L E S S O |
N 1.0 You will need ● patty paper Step 1 Step 2 Step 3 The Euler Line In the previous lessons you discovered the four points of concurrency: circumcenter, incenter, orthocenter, and centroid. In this activity you will discover how these points relate to a special line, the Euler line. The Euler line is named after the Swiss mathematician Leonhard Euler (1707–1783), who proved that three points of concurrency are collinear. Activity Three Out of Four You are going to look for a relationship among the points of concurrency. Draw a scalene triangle and have each person in your group trace the same triangle on a separate piece of patty paper. Have each group member construct with patty paper a different point of the four points of concurrency for the triangle. Record the group’s results by tracing and labeling all four points of concurrency on one of the four pieces of patty paper. What do you notice? Compare your group results with the results of other groups near you. State your discovery as a conjecture. Euler Line Conjecture The?,?, and? are the three points of concurrency that always lie on a line. The three special points that lie on the Euler line determine a segment called the Euler segment. The point of concurrency between the two endpoints of the Euler segment divides the segment into two smaller segments whose lengths have an exact ratio. EXPLORATION The Euler Line 189 Step 4 With a compass or patty paper, compare the lengths of the two parts of the Euler segment. What is the ratio? Compare your group’s results with the results of other groups and state your conjecture. Euler Segment Conjecture The? divides the Euler segment into two parts so that the smaller part is? the larger part. Step 5 Use your conjectures to solve this problem. AC is an Euler segment. AC 24 m. AB? BC? A BC IS THERE MORE TO THE ORTHOCENTER? At this point you may still wonder what’s special about the orthocenter. It does lie on the Euler line. Is there anything else surprising or special about it? Use geometry software to investigate the orthocenter. Draw a triangle ABC and construct its orthocenter O. Drag a vertex of the triangle around, and observe the behavior of the orthocenter. Where does the orthocenter lie in an acute triangle? |
An obtuse triangle? A right triangle? Drag the orthocenter. Describe how this affects the triangle. Hide the altitudes. Draw segments from each vertex to the orthocenter, as shown, forming three triangles within the original triangle. Now find the orthocenter of each of the three triangles. The Geometer’s Sketchpad was used to create this diagram and to hide the unnecessary lines. Using Sketchpad, you can quickly construct triangles and their points of concurrency. Once you make a conjecture, you can drag to change the shape of the triangle to see whether your conjecture is true. What happened? What does this mean? Experiment dragging different points, and observe the relationships among the four orthocenters. Drag the orthocenter toward each vertex. What happens? Write a paragraph about your findings, concluding with a conjecture about the orthocenter. Share your findings with your group members or classmates. B Orthocenter A C 190 CHAPTER 3 Using Tools of Geometry VIEW ● CHAPTER 11 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHA CHAPTER 3 R E V I E W In Chapter 1, you defined many terms that help establish the building blocks of geometry. In Chapter 2, you learned and practiced inductive reasoning skills. With the construction skills you learned in this chapter, you performed investigations that lay the foundation for geometry. The investigation section of your notebook should be a detailed report of the mathematics you’ve already done. Beginning in Chapter 1 and continuing in this chapter, you summarized your work in the definition list and the conjecture list. Before you begin the review exercises, make sure your conjecture list is complete. Do you understand each conjecture? Can you draw a clear diagram that demonstrates your understanding of each definition and conjecture? Can you explain them to others? Can you use them to solve geometry problems? EXERCISES For Exercises 1–10, identify the statement as true or false. For each false statement, explain why it is false, or sketch a counterexample. 1. In a geometric construction, you use a protractor and a ruler. 2. A diagonal is a line segment in a polygon that connects any two vertices. You will need Construction tools for Exercises 19–24 and 27–32 3. A trapezoid is a quadrilateral with exactly one pair of parallel sides. 4. A square is a rhombus with all angles congruent |
. 5. If a point is equidistant from the endpoints of a segment, then it must be the midpoint of the segment. 6. The set of all the points in the plane that are a given distance from a line segment is a pair of lines parallel to the given segment. 7. It is not possible for a trapezoid to have three congruent sides. 8. The incenter of a triangle is the point of intersection of the three angle bisectors. 9. The orthocenter of a triangle is the point of intersection of the three altitudes. 10. The incenter, the centroid, and the orthocenter are always inside the triangle. The Principles of Perspective, Italian, ca. 1780. Victoria and Albert Museum, London, Great Britain. CHAPTER 3 REVIEW 191 EW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTE For Exercises 11–18, match each geometric construction with one of the figures below. 11. Construction of a midsegment 12. Construction of an altitude 13. Construction of a centroid in a triangle 14. Construction of an incenter 15. Construction of an orthocenter in a triangle 16. Construction of a circumcenter 17. Construction of an equilateral triangle 18. Construction of an angle bisector A. D. G. J. B. E. H. K. C. F. I. L. Construction For Exercises 19–24, perform a construction with compass and straightedge or with patty paper. Choose the method for each problem, but do not mix the tools in any one problem. In other words, play each construction game fairly. 19. Draw an angle and construct a duplicate of it. 20. Draw a line segment and construct its perpendicular bisector. 192 CHAPTER 3 Using Tools of Geometry ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 21. Draw a line and a point not on the line. Construct a perpendicular to the line through the point. 22. Draw an angle and bisect it. 23. Construct an angle that measures 22.5°. 24. Draw a line and a point not on the line. Construct a second line so that it passes through the point and is parallel to the first line. 25. Brad and Janet are building a home for their pet hamsters, |
Riff and Raff, in the shape of a triangular prism. Which point of concurrency in the triangular base do they need to locate in order to construct the largest possible circular entrance? 26. Adventurer Dakota Davis has a map that once showed the location of a large bag of gold. Unfortunately, the part of the map that showed the precise location of the gold has burned away. Dakota visits the area shown on the map anyway, hoping to find clues. To his surprise, he finds three headstones with geometric symbols on them. The clues lead him to think that the treasure is buried at a point equidistant from the three stones. If Dakota’s theory is correct, how should he go about locating the point where the bag of gold might be buried? Construction For Exercises 27–32, use the given segments and angles to construct each figure. The lowercase letter above each segment represents the length of the segment. 27. ABC given A, C, and AC z 28. A segment with length 2y x 1 z 2 x y z C 29. PQR with PQ 3x, QR 4x, and PR 5x 30. Isosceles triangle ABD given A, and AB BD 2y A 31. Quadrilateral ABFD with mA mB, AD BF y, and AB 4x 32. Right triangle TRI with hypotenuse TI, TR x, and RI y and a square on TI, with TI as one side CHAPTER 3 REVIEW 193 EW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTE MIXED REVIEW Tell whether each symbol in Exercises 33–36 has reflectional symmetry, rotational symmetry, neither, or both. (The symbols are used in meteorology to show weather conditions.) 33. 34. 35. 36. For Exercises 37–40, match the term with its construction. 37. Centroid 38. Circumcenter 39. Incenter A. B. C. 40. Orthocenter D. For Exercises 41–54, identify the statement as true or false. For each false statement, explain why it is false or sketch a counterexample. 41. An isosceles right triangle is a triangle with an angle measuring 90° and no two sides congruent. 42. If two parallel lines are cut by a transversal, then the alternate interior angles are con |
gruent. 43. An altitude of a triangle must be inside the triangle. 44. The orthocenter of a triangle is the point of intersection of the three perpendicular bisectors of the sides. 45. If two lines are parallel to the same line, then they are parallel to each other. 46. If the sum of the measure of two angles is 180°, then the two angles are vertical angles. 47. Any two consecutive sides of a kite are congruent. 48. If a polygon has two pairs of parallel sides then it is a parallelogram. 49. The measure of an arc is equal to one half the measure of its central angle. 50. If TR is a median of TIE and point D is the centroid, then TD 3DR. 51. The shortest chord of a circle is the radius of a circle. 52. An obtuse triangle is a triangle that has one angle with measure greater than 90°. 194 CHAPTER 3 Using Tools of Geometry ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 53. Inductive reasoning is the process of showing that certain statements follow logically from accepted truths. 54. There are exactly three true statements in Exercises 41–54. 55. In the diagram, p q. a. Name a pair of corresponding angles. b. Name a pair of alternate exterior angles. c. If m3 42º, what is m6? 5 6 4 p q 2 3 1 In Exercises 56 and 57, use inductive reasoning to find the next number or shape in the pattern. 56. 100, 97, 91, 82, 70 57. Q 58. Consider the statement “If the month is October, then the month has 31 days.” a. Is the statement true? b. Write the converse of this statement. c. Is the converse true? 59. Find the point on the cushion at which a pool player should aim so that the white ball will hit the cushion and pass over point Q. For Exercises 60 and 61, find the function rule for the sequence. Then find the 20th term. 60. 61. 1 n f(n) 1 n f(n 15 5 11 5 24 6 … n … 20 14 … … 6 … n … 20 35 … … 62. Calculate each lettered angle measure. d 142° a c b g h 106° |
f 130° e CHAPTER 3 REVIEW 195 EW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTE 63. Draw a scalene triangle ABC. Use a 64. What’s wrong with this picture? straightedge and compass to construct the incenter of ABC. A 150° C 124° E F 56° B D 26° 65. What is the minimum number of regions that are formed by 100 distinct lines in a plane? What is the maximum number of regions formed by 100 lines in the plane? Assessing What You’ve Learned PERFORMANCE ASSESSMENT The subject of this chapter was the tools of geometry, so assessing what you’ve learned really means assessing what you can do with those tools. Can you do all the constructions you learned in this chapter? Can you show how you arrived at each conjecture? Demonstrating that you can do tasks like these is sometimes called performance assessment. Look over the constructions in the Chapter Review. Practice doing any of the constructions that you’re not absolutely sure of. Can you do each construction using either compass and straightedge or patty paper? Look over your conjecture list. Can you perform all the investigations that led to these conjectures? Demonstrate at least one construction and at least one investigation for a classmate, a family member, or your teacher. Do every step from start to finish, and explain what you’re doing. ORGANIZE YOUR NOTEBOOK Your notebook should have an investigation section, a definition list, and a conjecture list. Review the contents of these sections. Make sure they are complete, correct, and well organized. Write a one-page chapter summary from your notes. WRITE IN YOUR JOURNAL How does the way you are learning geometry—doing constructions, looking for patterns, and making conjectures—compare to the way you’ve learned math in the past? UPDATE YOUR PORTFOLIO Choose a construction problem from this chapter that you found particularly interesting and/or challenging. Describe each step, including how you figured out how to move on to the next step. Add this to your portfolio. 196 CHAPTER 3 Using Tools of Geometry CHAPTER 4 Discovering and Proving Triangle Properties Is it possible to make a representation of recognizable figures that has no background? M. C. ESCHER Symmetry Drawing E103, M. C. Escher, 1959 |
©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● learn why triangles are so useful in structures ● discover relationships between the sides and angles of triangles ● learn about the conditions that guarantee that two triangles are congruent L E S S O N 4.1 Teaching is the art of assisting discovery. ALBERT VAN DOREN Triangle Sum Conjecture Triangles have certain properties that make them useful in all kinds of structures, from bridges to high-rise buildings. One such property of triangles is their rigidity. If you build shelves like the first set shown at right, they will sway. But if you nail another board at the diagonal as in the second set, creating two triangles, you will have strong shelves. Rigid triangles such as these also give the bridge shown below its strength. Steel truss bridge over the Columbia River Gorge in Oregon Arranging congruent triangles can create parallel lines, as you can see from both the bridge and wood frame above. In this lesson you’ll discover that this property is related to the sum of the angle measures in a triangle. Architecture American architect Julia Morgan (1872–1957) designed many noteworthy buildings, including Hearst Castle in central California. She often used triangular trusses made of redwood and exposed beams for strength and openness, as in this church in Berkeley, California. This building is now the Julia Morgan Center for the Arts. 198 CHAPTER 4 Discovering and Proving Triangle Properties Investigation The Triangle Sum You will need ● a protractor ● a straightedge ● scissors ● patty paper There are an endless variety of triangles that you can draw, with different shapes and angle measures. Do their angle measures have anything in common? Start by drawing different kinds of triangles. Make sure your group has at least one acute and one obtuse triangle. Step 1 Step 2 Measure the three angles of each triangle as accurately as possible with your protractor. Find the sum of the measures of the three angles in each triangle. Compare results with others in your group. Does everyone get about the same result? What is it? Step 3 Check the sum another way. Write the letters a, b, and c in the interiors of the three angles of one of the triangles, and carefully cut out the triangle. c a b Step 4 Tear off the three angles. c a b Step 5 Arrange the three angles so that their vertices meet at a point. How does this |
arrangement show the sum of the angle measures? Compare results with others in your group. State a conjecture. c b a Triangle Sum Conjecture The sum of the measures of the angles in every triangle is?. C-17 LESSON 4.1 Triangle Sum Conjecture 199 Steps 1 through 5 may have convinced you that the Triangle Sum Conjecture is true, but a proof will explain why it is true for every triangle. Step 6 Copy and complete the paragraph proof below to explain the connection between the Parallel Lines Conjecture and the Triangle Sum Conjecture. Paragraph Proof: The Triangle Sum Conjecture To prove the Triangle Sum Conjecture, you need to show that the angle measures in a triangle add up to?. Start by drawing any ABC, and EC parallel to side AB. EC is called an auxiliary line, because it is an extra line that helps with the proof In the figure, m1 m2 m3 180° if you consider 1 2 as one angle whose measure is m1 m2, because?. You also know that EC AB, so m1 m4 and m3 m5, because?. So, by substituting for m1 and m3 in the first equation, you get?. Therefore, the measures of the angles in a triangle add up to?. Step 7 Suppose two angles of one triangle have the same measures as two angles of another triangle. What can you conclude about the third pair of angles? You can investigate Step 7 with patty paper. Draw a triangle on your paper. Create a second triangle on patty paper by tracing two of the angles of your original triangle, but make the side between your new angles a different length from the side between the angles you copied in the first triangle. How do the third angles in the two triangles compare? c a b x a b Step 8 Check your results with other students. You should be ready for your next conjecture. Third Angle Conjecture C-18 If two angles of one triangle are equal in measure to two angles of another triangle, then the third angle in each triangle?. 200 CHAPTER 4 Discovering and Proving Triangle Properties You can use the Triangle Sum Conjecture to show why the Third Angle Conjecture is true. You’ll do this in Exercise 15. EXAMPLE In the figure at right, is C congruent to E? Write a paragraph proof explaining why. A 60° B 70° Solution Yes. By the Third Angle Conjecture |
, because A and B are congruent to D and F, then C must be congruent to E. You could also use the Triangle Sum Conjecture to find that D and F both measure 50°. Since they have the same measure, they are congruent. C D 60° 70° F E EXERCISES 1. Technology Using geometry software, construct a triangle. Use the software to measure the three angles and calculate their sum. a. Drag the vertices and describe your observations. b. Repeat the process for a right triangle. You will need Geometry software for Exercise 1 Construction tools for Exercises 10–13 Use the Triangle Sum Conjecture to determine each lettered angle measure in Exercises 2–5. You might find it helpful to copy the diagrams so you can write on them. 2. x? 3. v? 4. z? 120° z 130° 100° v 6. Find the sum of the 7. Find the sum of the measures of the marked angles. measures of the marked angles. x 55° 52° 5. w? w 48° LESSON 4.1 Triangle Sum Conjecture 201 8 133° b 40° d 140° 47° 71° 9? 60° m 112° p 70° s t u q r n 40° In Exercises 10–12, use what you know to construct each figure. Use only a compass and a straightedge. A E R A L 10. Construction Given A and R of ARM, construct M. 11. Construction In LEG, mE mG. Given L, construct G. 12. Construction Given A, R, and side AE, construct EAR. 13. Construction Repeat Exercises 10–12 with patty-paper constructions. 14. In MAS below, M is a right angle. Let’s call the two acute angles, A and S, “wrong angles.” Write a paragraph proof or use algebra to show that “two wrongs make a right,” at least for angles in a right triangle. A M S 202 CHAPTER 4 Discovering and Proving Triangle Properties 15. Use the Triangle Sum Conjecture and the figures at right to write a paragraph proof explaining why the Third Angle Conjecture is true. 16. Write a paragraph proof, or use algebra, to explain why each angle of an equiangular triangle measures 60°. C x F y A B Review In Exercises 17 |
–21, tell whether the statement is true or false. For each false statement, explain why it is false or sketch a counterexample. D E 17. If two sides in one triangle are congruent to two sides in another triangle, then the two triangles are congruent. 18. If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are congruent. 19. If a side and an angle in one triangle are congruent to a side and an angle in another triangle, then the two triangles are congruent. 20. If three angles in one triangle are congruent to three angles in another triangle, then the two triangles are congruent. 21. If three sides in one triangle are congruent to three sides in another triangle, then the two triangles are congruent. 22. What is the number of stories in the tallest house you can build with two 52-card decks? How many cards would it take? One story (2 cards) Two stories (7 cards) Three stories (15 cards) IMPROVING YOUR VISUAL THINKING SKILLS Dissecting a Hexagon I Trace this regular hexagon twice. 1. Divide one hexagon into four congruent trapezoids. 2. Divide the other hexagon into eight congruent parts. What shape is each part? LESSON 4.1 Triangle Sum Conjecture 203 L E S S O N 4.2 Imagination is built upon knowledge. ELIZABETH STUART PHELPS Properties of Special Triangles Recall from Chapter 1 that an isosceles triangle is a triangle with at least two congruent sides. In an isosceles triangle, the angle between the two congruent sides is called the vertex angle, and the other two angles are called the base angles. The side between the two base angles is called the base of the isosceles triangle. The other two sides are called the legs. Vertex angle Legs Base angles Base In this lesson you’ll discover some properties of isosceles triangles. The Rock and Roll Hall of Fame and Museum structure is a pyramid containing many triangles that are isosceles and equilateral. The famous Transamerica Building in San Francisco contains many isosceles triangles. Architecture The Rock and Roll Hall of Fame and Museum in Cleveland, Ohio, is a dynamic structure. Its design reflects the innovative music that it |
honors. The front part of the museum is a large glass pyramid, divided into small triangular windows that resemble a Sierpin´ski tetrahedron, a three-dimensional Sierpin´ski triangle. The pyramid structure rests on a rectangular tower and a circular theater that looks like a performance drum. Architect I. M. Pei (b 1917) used geometric shapes to capture the resonance of rock and roll musical chords. 204 CHAPTER 4 Discovering and Proving Triangle Properties Investigation 1 Base Angles in an Isosceles Triangle Let’s examine the angles of an isosceles triangle. Each person in your group should draw a different angle for this investigation. Your group should have at least one acute angle and one obtuse angle. You will need ● patty paper ● a protractor C A B C Step 1 Step 2 C A Step 3 Step 1 Step 2 Step 3 Step 4 Draw an angle on patty paper. Label it C. This angle will be the vertex angle of your isosceles triangle. Place a point A on one ray. Fold your patty paper so that the two rays match up. Trace point A onto the other ray. Label the point on the other ray point B. Draw AB. You have constructed an isosceles triangle. Explain how you know it is isosceles. Name the base and the base angles. Use your protractor to compare the measures of the base angles. What relationship do you notice? How can you fold the paper to confirm your conclusion? Step 5 Compare results in your group. Was the relationship you noticed the same for each isosceles triangle? State your observations as your next conjecture. Isosceles Triangle Conjecture If a triangle is isosceles, then?. C-19 Equilateral triangles have at least two congruent sides, so they fit the definition of isosceles triangles. That means any properties you discover for isosceles triangles will also apply to equilateral triangles. How does the Isosceles Triangle Conjecture apply to equilateral triangles? You can switch the “if ” and “then” parts of the Isosceles Triangle Conjecture to obtain the converse of the conjecture. Is the converse of the Isosceles Triangle Conjecture true? Let’s investigate. LESSON 4.2 Properties of Special Triangles 205 You will need ● a compass ● a straightedge Investigation 2 Is |
the Converse True? Suppose a triangle has two congruent angles. Must the triangle be isosceles? A A C B Step 1 B Step 2 Step 1 Step 2 Step 3 Draw a segment and label it AB. Draw an acute angle at point A. This angle will be a base angle. (Why can’t you draw an obtuse angle as a base angle?) Copy A at point B on the same side of AB. Label the intersection of the two rays point C. Use your compass to compare the lengths of sides AC and BC. What relationship do you notice? How can you use patty paper to confirm your conclusion? A Step 4 Compare results in your group. State your observation as your next conjecture. B Converse of the Isosceles Triangle Conjecture If a triangle has two congruent angles, then?. C C-20 EXERCISES For Exercises 1–6, use your new conjectures to find the missing measures. 1. mH? 2. mG? You will need Construction tools for Exercises 12–14 3. mOLE? S G L E 35° O T 22° H O D 63° O 206 CHAPTER 4 Discovering and Proving Triangle Properties 4. mR? RM? A 68° M 68° 5. mY? RD? Y 35 cm R 7. Copy the figure at right. Calculate the measure of each lettered angle. R e 3.5 cm 28° D 6. The perimeter of MUD is 38 cm. mD? MD? M 14 cm 36° U D k h b a 56° 66° g f d c n p 8. The Islamic design below right is based on the star decagon construction shown below left. The ten angles surrounding the center are all congruent. Find the lettered angle measures. How many triangles are not isosceles? b e d c a 9. Study the triangles in the software constructions below. Each triangle has one vertex at the center of the circle, and two vertices on the circle. a. Are the triangles all isosceles? Write a paragraph proof explaining why or why not. b. If the vertex at the center of the first circle has an angle measure of 60°, find the measures of the other two angles in that triangle. LESSON 4.2 Properties of Special Triangles 207 Review In Exercises 10 and 11, complete the statement of con |
gruence from the information given. Remember to write the statement so that corresponding parts are in order. 10. GEA? E 24 cm G 36 cm A 36 cm N 24 cm C 11. JAN? N I 50° J A 40° E C In Exercises 12 and 13, use compass and straightedge, or patty paper, to construct a triangle that is not congruent to the given triangle, but has the given parts congruent. The symbol means “not congruent to.” 12. Construction Construct ABC DEF with A D, B E, and C F. 13. Construction Construct GHK MNP with HK NP, GH MN, and G M. D F E P N M 14. Construction With a straightedge and patty paper, construct an angle that measures 105°. In Exercises 15–18, determine whether each pair of lines through the points below is parallel, perpendicular, or neither. A(1, 3) B(6, 0) C(4, 3) D(1, 2) E(3, 8) F(4, 1) G(1, 6) H(4, 4) 15. AB and CD 16. FG and CD 17. AD and CH 18. DE and GH 19. Using the coordinate points above, is FGCD a trapezoid, a parallelogram, or neither? 20. Picture the isosceles triangle below toppling side over side to the right along the line. Copy the triangle and line onto your paper, then construct the path of point P through two cycles. Where on the number line will the vertex point land? P 0 4 8 16 24 208 CHAPTER 4 Discovering and Proving Triangle Properties For Exercises 21 and 22, use the ordered pair rule shown to relocate each of the vertices of the given triangle. Connect the three new points to create a new triangle. Is the new triangle congruent to the original one? Describe how the new triangle has changed position from the original. 21. (x, y) → (x 5, y 3) 22. (x, y) → (x, y) y (–1, 5) (1, 0) x (–3, 1) (3, 3) x y (–2, 3) (–3, –2) IMPROVING YOUR REASONING SKILLS Hundreds Puzzle Fill in the blanks of each equation below. All |
nine digits—1 through 9—must be used, in order! You may use any combination of signs for the four basic operations (,,, ), parentheses, decimal points, exponents, factorial signs, and square root symbols, and you may place the digits next to each other to create two-digit or three-digit numbers. Example: 1 2(3 4.5) 67 8 9 100 1. 1 2 3 4 5 6? 9 100 2. 1 2 3 4 5? 100 3. 1 2 [(3)(4)(5) 6]? 100 4. [(1? ) 5] 6 7 89 100 5. 1 23 4? 9 100 LESSON 4.2 Properties of Special Triangles 209 ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 4 ● USING YO USING YOUR ALGEBRA SKILLS 4 Writing Linear Equations A linear equation is an equation whose graph is a straight line. Linear equations are useful in science, business, and many other areas. For example, the linear 9c gives the rule for converting a temperature from degrees equation f 32 5 9 Celsius, c, to degrees Fahrenheit, f. The numbers 32 and 5 determine the graph of the equation. (100, 212) c 100° Understanding how the numbers in a linear equation determine the graph can help you write a linear equation based on information about a graph. The y-coordinate at which a graph crosses the y-axis is called the y-intercept. The measure of steepness is called the slope. Below are the graphs of four equations. The table gives the equation, slope, and y-intercept for each graph. How do the numbers in each equation relate to the slope and y-intercept? f 212° 32° (0, 32) y 9 y 3x 4 y 2 3x y 2x 1 x 6 3 _ y 5 x 2 Slope 3 Equation y 2 3x y 2x 1 2 y 3x -intercept 2 1 4 5 In each case, the slope of the line is the coefficient of x in the equation. The y-intercept is the constant that is added to, or subtracted from, the x term. In your algebra class, you may have learned about one of these forms of a linear equation in slope-intercept form: y a bx, where a is the y- |
intercept and b is the slope y mx b, where m is the slope and b is the y-intercept 210 CHAPTER 4 Discovering and Proving Triangle Properties ALGEBRA SKILLS 4 ● USING YOUR ALGEBRA SKILLS 4 ● USING YOUR ALGEBRA SKILLS 4 ● USING YO The only difference between these two forms is the order of the x term and the constant term. For example, the equation of a line with slope 3 and y-intercept 1 can be written as y 1 3x or y 3x 1. Let’s look at a few examples that show how you can apply what you have learned about the relationship between a linear equation and its graph. EXAMPLE A Find the equation of AB from its graph. Solution AB has y-intercept 2 and slope 3, so 4 the equation is y 2 3 x 4 EXAMPLE B Given points C(4, 6) and D(2, 3), find the equation of CD. Solution The slope of CD is, or 1 6 3. The slope 2 4 2 between any point (x, y) and one of the given points, say (4, 6), must also be EXAMPLE C Find the equation of the perpendicular bisector of the segment with endpoints (2, 9) and (6, 7). y Run 4 Rise 3 B (4, 1) x A (0, –2) y D (–2, 3) C (4, 6) y (2, 9) x x Solution The perpendicular bisector passes through the midpoint. The midpoint of the segment is 2 (7), 2 7 or (2, 1). The slope of the segment is 6 perpendicular is 1. Write the equation of its perpendicular bisector and solve 2 for y: 9, or 2. So the slope of its (6), 9 (–6, –7) y 1 x 2 USING YOUR ALGEBRA SKILLS 4 Writing Linear Equations 211 ALGEBRA SKILLS 4 ● USING YOUR ALGEBRA SKILLS 4 ● USING YOUR ALGEBRA SKILLS 4 ● USING YO EXERCISES In Exercises 1–3, graph each linear equation. 1. y 1 2x 2. y 4 x 4 3 3. 2y 3x 12 Write an equation for each line in Exercises 4 and 5. 4. y |
4 2 (0, 2) –2 –4 x 4 6 In Exercises 6–8, write an equation for the line through each pair of points. 5. y (–5, 8) (8, 2) x 6. (1, 2), (3, 4) 7. (1, 2), (3, 4) 8. (1, 2), (6, 4) 9. The math club is ordering printed T-shirts to sell for a fundraiser. The T-shirt company charges $80 for the set-up fee and $4 for each printed T-shirt. Using x for the number of shirts the club orders, write an equation for the total cost of the T-shirts. 10. Write an equation for the line with slope 3 that passes through the midpoint of the segment with endpoints (3, 4) and (11, 6). 11. Write an equation for the line that is perpendicular to the line y 4x 5 and that passes through the point (0, 3). For Exercises 12–14, the coordinates of the vertices of WHY are W(0, 0), H(8, 3), and Y(2, 9). 12. Find the equation of the line containing median WO. 13. Find the equation of the perpendicular bisector of side HY. 14. Find the equation of the line containing altitude HT. IMPROVING YOUR REASONING SKILLS Container Problem I You have an unmarked 9-liter container, an unmarked 4-liter container, and an unlimited supply of water. In table, symbol, or paragraph form, describe how you might end up with exactly 3 liters in one of the containers. 9 liters 4 liters 212 CHAPTER 4 Discovering and Proving Triangle Properties L E S S O N 4.3 Readers are plentiful, thinkers are rare. HARRIET MARTINEAU Drawbridges over the Chicago River in Chicago, Illinois Triangle Inequalities How long must each side of this drawbridge be so that the bridge spans the river when both sides come down? Triangles have similar requirements. In the triangles below, the blue segments are all congruent, and the red segments are all congruent. Yet, there are a variety of triangles. Notice how the length of the yellow segment changes. Notice also how the angle measures change. Can you form a triangle using sticks of any three lengths? How do the angle measures of a triangle relate to the lengths of |
its sides? In this lesson you will discover some geometric inequalities that answer these questions. LESSON 4.3 Triangle Inequalities 213 You will need ● a compass ● a straightedge Investigation 1 What Is the Shortest Path from A to B? Each person in your group should do each construction. Compare results when you finish. Step 1 Construct a triangle with each set of segments as sides. Given: C A C Construct: CAT Given Construct: FSH Were you able to construct CAT and FSH? Why or why not? Discuss your results with others. State your observations as your next conjecture. Step 2 Triangle Inequality Conjecture The sum of the lengths of any two sides of a triangle is? the length of the third side. C-21 The Triangle Inequality Conjecture relates the lengths of the three sides of a triangle. You can also think of it in another way: The shortest path between two points is along the segment connecting them. In other words, the path from A to C to B can’t be shorter than the path from A to B. AB = 3 cm AC + CB = 4 cm C AB = 3 cm AC + CB = 3.5 cm AB = 3 cm AC + CB = 3.1 cm AB = 3 cm AC + CB = 3 cm You can use geometry software to compare two different paths. 214 CHAPTER 4 Discovering and Proving Triangle Properties You will need ● a ruler ● a protractor Investigation 2 Where Are the Largest and Smallest Angles? Each person should draw a different scalene triangle for this investigation. Some group members should draw acute triangles, and some should draw obtuse triangles. L Step 1 Step 2 Step 3 Measure the angles in your triangle. Label the angle with greatest measure L, the angle with second greatest measure M, and the smallest angle S. Measure the three sides. Label the longest side l, the second longest side m, and the shortest side s. Which side is opposite L? M? S? M M L S S Discuss your results with others. Write a conjecture that states where the largest and smallest angles are in a triangle, in relation to the longest and shortest sides. Side-Angle Inequality Conjecture C-22 In a triangle, if one side is longer than another side, then the angle opposite the longer side is?. Exterior angle Adjacent interior angle Remote interior angles So far in this chapter, you have studied interior angles of triangles. Triangles also have |
exterior angles. If you extend one side of a triangle beyond its vertex, then you have constructed an exterior angle at that vertex. Each exterior angle of a triangle has an adjacent interior angle and a pair of remote interior angles. The remote interior angles are the two angles in the triangle that do not share a vertex with the exterior angle. You will need ● a straightedge ● patty paper Step 1 Investigation 3 Exterior Angles of a Triangle Each person should draw a different scalene triangle for this investigation. Some group members should draw acute triangles, and some should draw obtuse triangles. On your paper, draw a scalene triangle, ABC. Extend AB beyond point B and label a point D outside the triangle on AB. Label the angles as shown. C c a A b x DB LESSON 4.3 Triangle Inequalities 215 Step 2 Step 3 Step 4 Copy the two remote interior angles, A and C, onto patty paper to show their sum. How does the sum of a and c compare with x? Use your patty paper from Step 2 to compare. c a Discuss your results with your group. State your observations as a conjecture. Triangle Exterior Angle Conjecture The measure of an exterior angle of a triangle?. C-23 You just discovered the Triangle Exterior Angle Conjecture by inductive reasoning. You can use the Triangle Sum Conjecture, some algebra, and deductive reasoning to show why the Triangle Exterior Angle Conjecture is true for all triangles. You’ll do the paragraph proof on your own in Exercise 17. EXERCISES In Exercises 1–4, determine whether it is possible to draw a triangle with sides of the given measures. If possible, write yes. If not possible, write no and make a sketch demonstrating why it is not possible. 1. 3 cm, 4 cm, 5 cm 2. 4 m, 5 m, 9 m 3. 5 ft, 6 ft, 12 ft 4. 3.5 cm, 4.5 cm, 7 cm In Exercises 5–10, use your new conjectures to arrange the unknown measures in order from greatest to least. 5. c 70° 8. 17 in. b 35° b a a 6. 9. 55° c a b 68° 15 in. c 28 in. 30° a 9 cm c 12 cm 7. 5 cm b a 10. b 72° c z 28° 34° w y v x 42° 30° 11. |
If 54 and 48 are the lengths of two sides of a triangle, what is the range of possible values for the length of the third side? 216 CHAPTER 4 Discovering and Proving Triangle Properties 12. What’s wrong with this picture? Explain. 13. What’s wrong with this picture? Explain. 11 cm 25 cm 48 cm 72° 72° 74° 74° In Exercises 14–16, use one of your new conjectures to find the missing measures. 14. t p? 15. r? 16. x? p r 135° t 130° 58° 17. Use algebra and the Triangle Sum Conjecture to explain why the Triangle Exterior Angle Conjecture is true. Use the figure at right. 18. Read the Recreation Connection below. If you want to know the perpendicular distance from a landmark to the path of your boat, what should be the measurement of your bow angle when you begin recording? a A x B b 144° c x C D Recreation Geometry is used quite often in sailing. For example, to find the distance between the boat and a landmark on shore, sailors use a rule called doubling the angle on the bow. The rule says, measure the angle on the bow (the angle formed by your path and your line of sight to the landmark) at point A. Check your bearing until, at point B, the bearing is double the reading at point A. The distance traveled from A to B is also the distance from the landmark to your new position. L A B LESSON 4.3 Triangle Inequalities 217 Review In Exercises 19 and 20, calculate each lettered angle measure. 20. 19. d c a b 32° 38° f e d g h c b a 22° In Exercises 21–23, complete the statement of congruence. 21. BAR? 22. FAR? E A B R N K 52° A R 38° F 23. HG HJ HEJ? G J E O H RANDOM TRIANGLES Imagine you cut a 20 cm straw in two randomly selected places anywhere along its length. What is the probability that the three pieces will form a triangle? How do the locations of the cuts affect whether or not the pieces will form a triangle? Explore this situation by cutting a straw in different ways, or use geometry software to model different possibilities. Based on your informal exploration, predict the probability of the pieces forming a triangle. Now generate a large number of randomly chosen |
lengths to simulate the cutting of the straw. Analyze the results and calculate the probability based on your data. How close was your prediction? Your project should include Your prediction and an explanation of how you arrived at it. Your randomly generated data. An analysis of the results and your calculated probability. An explanation of how the location of the cuts affects the chances of a triangle being formed. You can use Fathom to generate many sets of random numbers quickly. You can also set up tables to view your data, and enter formulas to calculate quantities based on your data. 218 CHAPTER 4 Discovering and Proving Triangle Properties L E S S O N 4.4 The person who knows how will always have a job; the person who knows why will always be that person’s boss. ANONYMOUS Are There Congruence Shortcuts? A building contractor has just assembled two massive triangular trusses to support the roof of a recreation hall. Before the crane hoists them into place, the contractor needs to verify that the two triangular trusses are identical. Must the contractor measure and compare all six parts of both triangles? You learned from the Third Angle Conjecture that if there is a pair of angles congruent in each of two triangles, then the third angles must be congruent. But will this guarantee that the trusses are the same size? You probably need to also know something about the sides in order to be sure that two triangles are congruent. Recall from earlier exercises that fewer than three parts of one triangle can be congruent to corresponding parts of another triangle, without the triangles being congruent. So let’s begin looking for congruence shortcuts by comparing three parts of each triangle. There are six different ways that the same three parts of two triangles may be congruent. They are diagrammed below. An angle that is included between two sides of a triangle is called an included angle. A side that is included between two angles of a triangle is called an included side. Side-Side-Side (SSS) Side-Angle-Side (SAS) Angle-Side-Angle (ASA) Three pairs of congruent sides Two pairs of congruent sides and one pair of congruent angles (angles between the pairs of sides) Two pairs of congruent angles and one pair of congruent sides (sides between the pairs of angles) Side-Angle-Angle (SAA) Side-Side |
-Angle (SSA) Angle-Angle-Angle (AAA) Two pairs of congruent angles and one pair of congruent sides (sides not between the pairs of angles) Two pairs of congruent sides and one pair of congruent angles (angles not between the pairs of sides) Three pairs of congruent angles LESSON 4.4 Are There Congruence Shortcuts? 219 You will need ● a compass ● a straightedge You will consider three of these cases in this lesson and three others in the next lesson. Let’s begin by investigating SSS and SAS. Investigation 1 Is SSS a Congruence Shortcut? First you will investigate the Side-Side-Side (SSS) case. If the three sides of one triangle are congruent to the three sides of another, must the two triangles be congruent? Step 1 Step 2 Construct a triangle from the three parts shown. Be sure you match up the endpoints labeled with the same letter. A B A C C B Compare your triangle with the triangles made by others in your group. (One way to compare them is to place the triangles on top of each other and see if they coincide.) Is it possible to construct different triangles from the same three parts, or will all the triangles be congruent? Step 3 You are now ready to complete the conjecture for the SSS case. SSS Congruence Conjecture C-24 If the three sides of one triangle are congruent to the three sides of another triangle, then?. Career Congruence is very important in design and manufacturing. Modern assembly-line production relies on identical, or congruent, parts that are interchangeable. In the assembly of an automobile, for example, the same part needs to fit into each car coming down the assembly line. 220 CHAPTER 4 Discovering and Proving Triangle Properties You will need ● a compass ● a straightedge Step 1 Step 2 Investigation 2 Is SAS a Congruence Shortcut? Next you will consider the Side-Angle-Side (SAS) case. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, must the triangles be congruent? Construct a triangle from the three parts shown. Be sure you match up the endpoints labeled with the same letter. D D E F Compare your triangle with the triangles made by others in your group. (One way to compare them is |
to place the triangles on top of each other and see if they coincide.) Is it possible to construct different triangles from the same three parts, or will all the triangles be congruent? D Step 3 You are now ready to complete the conjecture for the SAS case. SAS Congruence Conjecture C-25 If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then?. Next, let’s look at the Side-Side-Angle (SSA) case. EXAMPLE If two sides and a non-included angle of one triangle are congruent to two corresponding sides and a non-included angle of another, must the triangles be congruent? In other words, can you construct only one triangle with the two sides and a non-included angle shown below? S T U T Solution Once you construct ST on a side of S, there are two possible locations for point U on the other side of the angle. Point U can be here or here. U S U S T LESSON 4.4 Are There Congruence Shortcuts? 221 So two different triangles are possible in the SSA case, and the triangles are not necessarily congruent. U U S T S T There is a counterexample for the SSA case, so it is not a congruence shortcut. EXERCISES For Exercises 1–6, decide whether the triangles are congruent, and name the congruence shortcut you used. If the triangles cannot be shown to be congruent as labeled, write “cannot be determined.” You will need Construction tools for Exercises 18 and 19 1. Which conjecture tells you LUZ IDA? 2. Which conjecture tells you AFD EFD? 3. Which conjecture tells you COT NPA? L 65° Z D U A 65° I F A E D 4. Which conjecture tells you CAV CEV? 5. Which conjecture tells you KAP AKQ. Y is a midpoint. Which conjecture tells you AYB RYN? B N A Y R 7. Explain why the boards that are nailed diagonally in the corners of this wooden gate make the gate stronger and prevent it from changing its shape under stress. 8. What’s wrong with this picture? 130° a b 125° 222 CHAPTER 4 Discovering and Proving Triangle Properties In Exercises |
9–14, name a triangle congruent to the given triangle and state the congruence conjecture. If you cannot show any triangles to be congruent from the information given, write “cannot be determined” and explain why. 9. ANT? N L T A E F 12. MAN? N Y M A B O 10. RED? B R 11. WOM? O E D U L M W T 13. SAT? O A S T 14. GIT? T G I A N In Exercises 15 and 16, determine whether the segments or triangles in the coordinate plane are congruent and explain your reasoning. 15. SUN? y 6 U S R A –6 –6 Y N x 6 16. DRO? y 6 4 2 P S –6 –4 D O 2 4 6 x 17. NASA scientists using a lunar exploration vehicle (LEV) wish to determine the distance across the deep crater shown at right. They have mapped out a path for the LEV as shown. How can the scientists use this set of measurements to calculate the approximate diameter of the crater? In Exercises 18 and 19, use a compass and straightedge, or patty paper, to perform these constructions. 18. Construction Draw a triangle. Use the SSS Congruence Conjecture to construct a second triangle congruent to the first. 19. Construction Draw a triangle. Use the SAS Congruence Conjecture to construct a second triangle congruent to the first. R –4 –6 Finish x Start LESSON 4.4 Are There Congruence Shortcuts? 223 Review 20. Copy the figure. Calculate the measure of each lettered angle. g m p r s b 143° a 85° d c e hk f 48° 21. If two sides of a triangle measure 8 cm and 11 cm, what is the range of values for the length of the third side? 22. How many “elbow,” “T,” and “cross” pieces do you need to build a 20-by-20 grid? Start with the smaller grids shown below. Copy and complete the table. Elbow : T: Cross: 3 4 5 … n … 20 Side length Elbows T’s Crosses 1 4 0 0 2 4 4 1 23. Find the point of intersection of the lines y 2 x 1 and 3x 4y 8. 3 24. Isos |
celes right triangle ABC has vertices with coordinates A(8, 2), B(5, 3), and C(0, 0). Find the coordinates of the orthocenter. IMPROVING YOUR REASONING SKILLS Container Problem II You have a small cylindrical measuring glass with a maximum capacity of 250 mL. All the marks have worn off except the 150 mL and 50 mL marks. You also have a large unmarked container. It is possible to fill the large container with exactly 350 mL. How? What is the fewest number of steps required to obtain 350 mL? 250 150 50 224 CHAPTER 4 Discovering and Proving Triangle Properties L E S S O N 4.5 There is no more a math mind, than there is a history or an English mind. GLORIA STEINEM You will need ● a compass ● a straightedge Step 1 Step 2 Are There Other Congruence Shortcuts? In the last lesson, you discovered that there are six ways that three parts of two triangles can be the same. You found that SSS and SAS both lead to the congruence of the two triangles, but that SSA does not. Is the Angle-Angle-Angle (AAA) case a congruence shortcut? You may recall exercises that explored the AAA case. For example, these triangles have three congruent angles, but they do not have congruent sides. M There is a counterexample for the AAA case, so it is not a congruence shortcut. Next, let’s investigate the ASA case. P N Q O R Investigation Is ASA a Congruence Shortcut? Consider the Angle-Side-Angle (ASA) case. If two angles and the included side of one triangle are congruent to two angles and the included side of another, must the triangles be congruent? Construct a triangle from the three parts shown. Be sure that the side is included between the given angles. M T M T Compare your triangle with the triangles made by others in your group. Is it possible to construct different triangles from the same three parts, or will all the triangles be congruent? Step 3 You are now ready to complete the conjecture for the ASA case. ASA Congruence Conjecture C-26 If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then?. Let’s assume that the SSS, SAS, |
and ASA Congruence Conjectures are true for all pairs of triangles that have those sets of corresponding parts congruent. LESSON 4.5 Are There Other Congruence Shortcuts? 225 The ASA case is closely related to another special case—the Side-Angle-Angle (SAA) case. You can investigate the SAA case with compass and straightedge, but you will have to use trialand-error to accurately locate the second angle vertex because the side that is given is not the included side JK is too short. JK is too long. JK is the right length. EXAMPLE Solution If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another, must the triangles be congruent? Let’s look at it deductively. In triangles ABC and XYZ, A X, B Y, and BC YZ. Is ABC XYZ? Explain your answer in a paragraph. Two angles in one triangle are congruent to two angles in another. The Third Angle Conjecture says that C Z. The diagram now looks like this So you now have two angles and the included side of one triangle congruent to two angles and the included side of another. By the ASA Congruence Conjecture, ABC XYZ. So the SAA Congruence Conjecture follows easily from the ASA Congruence Conjecture. Complete the conjecture for the SAA case. SAA Congruence Conjecture C-27 If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then?. 226 CHAPTER 4 Discovering and Proving Triangle Properties Four of the six cases—SSS, SAS, ASA, and SAA—turned out to be congruence shortcuts. The diagram for each case is shown below. SAS SAA SSS ASA Add these diagrams, along with your congruence shortcut conjectures, to your conjecture list. Many structures use congruent triangles for symmetry and strength. Can you tell which triangles in this toy structure are congruent? EXERCISES For Exercises 1–6, determine whether the triangles are congruent, and name the congruence shortcut. If the triangles cannot be shown to be congruent, write “cannot be determined.” You will need Construction tools for Exerc |
ises 17–19, 20, and 23 1. AMD RMC D A M C R 2. BOX CAR 3. GAS IOL. HOW FEW 5. FSH FSI 6. ALT INT LESSON 4.5 Are There Other Congruence Shortcuts? 227 In Exercises 7–14, name a triangle congruent to the triangle given and state the congruence conjecture. If you cannot show any triangles to be congruent from the information given, write “cannot be determined” and explain why. 7. FAD? F A E D 10. PO PR POE? SON? O N P S E R 8. OH AT WHO? A T W H O 11.?? D A M C R 13. BLA? 14. LAW? B L A C K W K A L 16. Use slope properties to show AB BC, CD DA, and BC DA. ABC?. Why? In Exercises 17–19, use a compass and a straightedge, or patty paper, to perform each construction. 17. Construction Draw a triangle. Use the ASA Congruence Conjecture to construct a second triangle congruent to the first. Write a paragraph to justify your steps. 18. Construction Draw a triangle. Use the SAA Congruence Conjecture to construct a second triangle congruent to the first. Write a paragraph to justify your method. 9. AT is an angle bisector. LAT? L T S 12. RMF? F M A A R 15. SLN is equilateral. Is TIE equilateral? Explain. N E T S A –6 19. Construction Construct two triangles that are not congruent, even though the three angles of one triangle are congruent to the three angles of the other. 228 CHAPTER 4 Discovering and Proving Triangle Properties Review 20. Construction Using only a compass and a straightedge, construct an isosceles triangle with a vertex angle that measures 135°. 21. If n concurrent lines divide the plane into 250 parts then n?. 22. “If the two diagonals of a quadrilateral are perpendicular, then the quadrilateral is a rhombus.” Explain why this statement is true or sketch a counterexample. 23. Construction Construct an isosceles right triangle with KM as one of the legs. How many noncongruent triangles can you construct? Why |
? K M 24. Sketch five lines in a plane that intersect in exactly five points. Now do this in a different way. 25. APPLICATION Scientists use seismograms and a method called triangulation to pinpoint the epicenter of an earthquake. a. Data recorded for one quake show that the epicenter is 480 km from Eureka, California; 720 km from Elko, Nevada; and 640 km from Las Vegas, Nevada. Trace the locations of these three towns and use the scale and your construction tools to find the location of the epicenter. b. Is it necessary to have seismogram information from three towns? Would two towns suffice? Explain. OREGON Eureka Boise IDAHO Sacramento San Francisco Reno Elko NEVADA Las Vegas UTAH P-wave arrival time S-wave arrival time S-P interval Los Angeles Miles 0 50 100 200 0 100 Kilometers 400 ARIZONA Time in seconds 250 200 150 100 50 0 mm 50 100 150 200 250 e d u t i l p m A IMPROVING YOUR ALGEBRA SKILLS Algebraic Sequences I Find the next two terms of each algebraic sequence. x 3y, 2x y, 3x 4y, 5x 5y, 8x 9y, 13x 14y,?,? x 7y, 2x 2y, 4x 3y, 8x 8y, 16x 13y, 32x 18y,?,? LESSON 4.5 Are There Other Congruence Shortcuts? 229 L E S S O N 4.6 The job of the younger generation is to find solutions to the solutions found by the older generation. ANONYMOUS Corresponding Parts of Congruent Triangles In Lessons 4.4 and 4.5, you discovered four shortcuts for showing that two triangles are congruent—SSS, SAS, ASA, and SAA. The definition of congruent triangles states that if two triangles are congruent, then the corresponding parts of those congruent triangles are congruent. We’ll use the letters CPCTC to refer to the definition. Let’s see how you can use congruent triangles and CPCTC. C B M 2 1 A D EXAMPLE A Is AD BC in the figure above? Use a deductive argument to explain why they must be congruent. Solution Here is one possible explanation: 1 2 because they are vertical angles. And |
it is given that AM BM and A B. So, by ASA, AMD BMC. Because the triangles are congruent, AD BC by CPCTC. If you use a congruence shortcut to show that two triangles are congruent, then you can use CPCTC to show that any of their corresponding parts are congruent. When you are trying to prove that triangles are congruent, it can be hard to keep track of what you know. Mark all the information on the figure. If the triangles are hard to see, use different colors or redraw them separately. EXAMPLE B Is AE BD? Write a paragraph proof explaining why. D C F E A B Solution The triangles you can use to show congruence are ABD and BAE. You can separate or color them to see them more clearly Separated triangles B D C F E A B Color-coded triangles 230 CHAPTER 4 Discovering and Proving Triangle Properties You can see that the two triangles have two pairs of congruent angles and they share a side. Paragraph Proof: Show that AE BD. In ABD and BAE, D E and B A. Also, AB BA because they are the same segment. So ABD BAE by SAA. By CPCTC, AE BD. EXERCISES You will need For Exercises 1–9, copy the figures onto your paper and mark them with the given information. Answer the question about segment or angle congruence. If your answer is yes, write a paragraph proof explaining why. Remember to state which congruence shortcut you used. If there is not enough information to prove congruence, write “cannot be determined.” Construction tools for Exercises 16 and 17 1. A C, ABD CBD Is AB CB? B A C D 4. S I, G A T is the midpoint of SI. Is SG IA. BT EU, BU ET Is B E? B E R 2. CN WN, C W Is RN ON? 3. CS HR, 1 2 Is CR HS. FO FR, UO UR Is O R? 6. MN MA, ME MR Is E R. HALF is a parallelogram. Is HA HF? H 9. D C, O A, G T. Is TA GO LESSON 4.6 Corresponding Parts of Congruent Triangles 231 For Exercises 10 and 11, you can use the right angles |
and the lengths of horizontal and vertical segments shown on the grid. Answer the question about segment or angle congruence. If your answer is yes, explain why. 10. Is FR GT? Why? 11. Is OND OCR? Why? T R O G E F 12. In Chapter 3, you used inductive reasoning to discover how to duplicate an angle using a compass and straightedge. Now you have the skills to explain why the construction works using deductive reasoning. The construction is shown at right. Write a paragraph proof explaining why it works. B Review y 6 D –6 C N O R x 6 –6 A C E D F In Exercises 13–15, complete each statement. If the figure does not give you enough information to show that the triangles are congruent, write “cannot be determined.” 13. AM is a median. CAM? 14. HEI? Why? 15. U is the midpoint of both FE and LT. ULF? 16. Construction Draw a triangle. Use the SAS Congruence Conjecture to construct a second triangle congruent to the first. 17. Construction Construct two triangles that are not congruent, even though two sides and a non-included angle of one triangle are congruent to two sides and a corresponding non-included angle of the other triangle. 18. Copy the figure. Calculate the measure of each lettered angle. 232 CHAPTER 4 Discovering and Proving Triangle Properties c b k h l a 68° d e m f g 19. According to math legend, the Greek mathematician Thales (ca. 625–547 B.C.E.) could tell how far out to sea a ship was by using congruent triangles. First, he marked off a long segment in the sand. Then, from each endpoint of the segment, he drew the angle to the ship. He then remeasured the two angles on the other side of the segment away from the shore. The point where the rays of these two angles crossed located the ship. What congruence conjecture was Thales using? Explain. 20. Isosceles right triangle ABC has vertices A(8, 2), B(5, 3), and C(0, 0). Find the coordinates of the circumcenter. 21. The SSS Congruence Conjecture explains why triangles are rigid structures though other polygons are not. By adding one “strut” |
(diagonal) to a quadrilateral you create a quadrilateral that consists of two triangles, and that makes it rigid. What is the minimum number of struts needed to make a pentagon rigid? A hexagon? A dodecagon? What is the minimum number of struts needed to make other polygons rigid? Complete the table and make your conjecture.?? Number of sides 3 4 5 6 7 … 12 … n Number of struts needed to make polygon rigid … … 22. Line is parallel to AB. If P moves to the right along, which of the following always decreases? a. The distance PC b. The distance from C to AB c. The ratio A B P A d. AC AP P C A B 23. Find the lengths x and y. Each angle is a right angle LESSON 4.6 Corresponding Parts of Congruent Triangles 233 POLYA’S PROBLEM George Polya (1887–1985) was a mathematician who specialized in problem-solving methods. He taught mathematics and problem solving at Stanford University for many years, and wrote the book How to Solve It. He posed this problem to his students: Into how many parts will five random planes divide space? 1 plane 2 planes 3 planes It is difficult to visualize five random planes intersecting in space. What strategies would you use to find the answer? Your project is to solve this problem, and to show how you know your answer is correct. Here are some of Polya’s problem-solving strategies to help you. Understand the problem. Draw a figure or build a model. Can you restate the problem in your own words? Break down the problem. Have you done any simpler problems that are like this one? 1 line 2 lines 3 lines 1 point 2 points 3 points Check your answer. Can you find the answer in a different way to show that it is correct? (The answer, by the way, is not 32!) Your method is as important as your answer. Keep track of all the different things you try. Write down your strategies, your results, and your thinking, as well as your answer. 234 CHAPTER 4 Discovering and Proving Triangle Properties L E S S O N 4.7 Flowchart Thinking You have been making many discoveries about triangles. As you try to explain why the new conjectures are true, you build upon definitions and conjectures you made before. If you can only find it, there is a |
reason for everything. TRADITIONAL SAYING So far, you have written your explanations as paragraph proofs. First, we’ll look at a diagram and explain why two angles must be congruent, by writing a paragraph proof, in Example A. Then we’ll look at a different tool for writing proofs, and use that tool to write the same proof, in Example B. EXAMPLE A In the figure at right, EC AC and ER AR. Is A E? If so, give a logical argument to explain why they are congruent. R R E A E A C C First mark the given information on the figure. Then consider whether A is congruent to E, and why. Paragraph Proof: Show that A E. EC AC and ER AR because that information is given. RC RC because it is the same segment, and any segment is congruent to itself. So, CRE CRA by the SSS Congruence Conjecture. If CRE CRA, then A E by CPCTC. Were you able to follow the logical steps in Example A? Sometimes a logical argument or a proof is long and complex, and a paragraph might not be the clearest way to present all the steps. In Chapter 1, you used concept maps to visualize the relationships among different kinds of polygons. A flowchart is a concept map that shows all the steps in a complicated procedure in proper order. Arrows connect the boxes to show how facts lead to conclusions. Flowcharts make your logic visible so that others can follow your reasoning. To present your reasoning in flowchart form, create a flowchart proof. Place each statement in a box. Write the logical reason for each statement beneath its box. For example, you would write “RC RC, because it is the same segment,” as RC RC Same segment LESSON 4.7 Flowchart Thinking 235 Solution Career If section-num-In section-num-comp No Yes Write major-tot-line Write minor-tot-line Add 2 To line-ct C Computer programmers use programming language and detailed plans to design computer software. They often use flowcharts to plan the logic in programs. Here is the same logical argument that you created in Example A in flowchart proof format. EXAMPLE B In the figure below, EC AC and ER AR. Is E A? If so, write a flowchart proof to explain why. R E A C Solution First, restate the given information |
clearly. It helps to mark the given information on the figure. Then state what you are trying to show. Given: AR ER EC AC Show: E A Flowchart Proof C R E 1 AR ER Given 2 3 EC AC Given RC RC Same segment A 4 RCE RCA 5 E A SSS Congruence Conjecture CPCTC In a flowchart proof, the arrows show how the logical argument flows from the information that is given to the conclusion that you are trying to prove. Drawing an arrow is like saying “therefore.” You can draw flowcharts top to bottom or left to right. Compare the paragraph proof in Example A with the flowchart proof in Example B. What similarities and differences are there? What are the advantages of each format? Is this contraption like a flowchart proof? 236 CHAPTER 4 Discovering and Proving Triangle Properties EXERCISES 1. Suppose you saw this step in a proof: Construct angle bisector CD to the midpoint of side AB in ABC. What’s wrong with that step? Explain. 2. Copy the flowchart. Provide each missing reason or statement in the proof. Given: SE SU E U Show: MS OS Flowchart Proof 1 SE SU? MS SO ASA Congruence Conjecture? C D B A O 3. Copy the flowchart. Provide each missing reason or statement in the proof. Given: I is the midpoint of CM I is the midpoint of BL Show: CL MB Flowchart Proof C L 2 1 I B M 1 I is midpoint of CM 2 I is midpoint of BL Given? 3 CI IM Definition of midpoint 4 IL IB? 5 1 2? 6??? 7? CPCTC LESSON 4.7 Flowchart Thinking 237 In Exercises 4–6, an auxiliary line segment has been added to the figure. 4. Complete this flowchart proof of the Isosceles Triangle Conjecture. Given that the triangle is isosceles, show that the base angles are congruent. Given: NEW is isosceles, with WN EN and median NS Show: W E Flowchart Proof 1 NS is a median 3 NS NS Given Same segment N S W 2 S is a midpoint Definition of median 4 5 WS SE Definition of midpoint WN NE? 6 WSN?? 7 W?? 5. Complete this flowchart proof of the Converse of the Isosceles Triangle Conject |
ure. Given: NEW with W E NS is an angle bisector Show: NEW is an isosceles triangle Flowchart Proof 1 NS is an angle bisector Given 2 1? Definition of? WN NE??? 7 NEW is isosceles? 4 NS NS? 6. Complete the flowchart proof. What does this proof tell you about parallelograms? Given: SA NE SE NA Show: SA NE Flowchart Proof E N 3 2 1 SA NE? 2 SE NA? 3 3 4 AIA Conjecture SN SN Same segment 238 CHAPTER 4 Discovering and Proving Triangle Properties 7. In Chapter 3, you learned how to construct the bisector of an angle. Now you have the skills to explain why the construction works, using deductive reasoning. Create a paragraph or flowchart proof to show that the construction method works. Given: ABC with BA BC, CD AD Show: BD is the angle bisector of ABC B 1 2 A C D Review 8. Which segment is the shortest? Explain. 9. What’s wrong with this picture? Explain. E 30° 61° N 59° 121° A 29° L 44° 42° For Exercises 10–12, name the congruent triangles and explain why the triangles are congruent. If you cannot show that they are congruent, write “cannot be determined.” 10. PO PR 11.?? 12. AC CR, CK is a median of ARC. RCK? POE? SON? 13. Copy the figure below. Calculate the measure of each lettered angle. C 1 2 1 k m 2 and 72° f 14. Which point of concurrency is equidistant from all three vertices? Explain why. Which point of concurrency is equidistant from all three sides? Explain why. LESSON 4.7 Flowchart Thinking 239 15. Samantha is standing at the bank of a stream, wondering how wide the stream is. Remembering her geometry conjectures, she kneels down and holds her fishing pole perpendicular to the ground in front of her. She adjusts her hand on the pole so that she can see the opposite bank of the stream along her line of sight through her hand. She then turns, keeping a firm grip on the pole, and uses the same line of sight to spot a boulder on her side of the stream. She measures the distance to the boulder and concludes that this equals the distance across the stream. |
What triangle congruence shortcut is Samantha using? Explain. 16. What is the probability of randomly selecting one of the shortest diagonals from all the diagonals in a regular decagon? 17. Sketch the solid shown with the red 18. Sketch the new location of rectangle BOXY and green cubes removed. after it has been rotated 90° clockwise about the origin. y 4 Y B X O x 4 IMPROVING YOUR REASONING SKILLS Pick a Card Nine cards are arranged in a 3-by-3 array. Every jack borders on a king and on a queen. Every king borders on an ace. Every queen borders on a king and on an ace. (The cards border each other edge-to-edge, but not corner-to-corner.) There are at least two aces, two kings, two queens, and two jacks. Which card is in the center position of the 3-by-3 array? 240 CHAPTER 4 Discovering and Proving Triangle Properties L E S S O N 4.8 The right angle from which to approach any problem is the try angle. ANONYMOUS Proving Isosceles Triangle Conjectures This boathouse is a remarkably symmetric structure with its isosceles triangle roof and the identical doors on each side. The rhombus-shaped attic window is centered on the line of symmetry of this face of the building. What might this building reveal about the special properties of the line of symmetry in an isosceles triangle? In this lesson you will make a conjecture about a special segment in isosceles triangles. Then you will use logical reasoning to prove your conjecture is true for all isosceles triangles. First, consider a scalene triangle. In ARC, CD is the altitude to the base AR, CE is the angle bisector of ACR, and CF is the median to the base AR. From this example it is clear that the angle bisector, the altitude, and the median can all be different line segments. Is this true for all triangles? Can two of these ever be the same segment? Can they all be the same segment? Let’s investigate. C A F E D R LESSON 4.8 Proving Isosceles Triangle Conjectures 241 Investigation The Symmetry Line in an Isosceles Triangle Each person in your group should draw a different isosceles triangle for this investigation. You will need ● a compass ● a |
straightedge Step 1 Step 2 Step 3 Step 4 Step 5 K Construct a large isosceles triangle on a sheet of unlined paper. Label it ARK, with K the vertex angle. Construct angle bisector KD with point D on AR. Do ADK and RDK look congruent? With your compass, compare AD and RD. Is D the midpoint of AR? If D is the midpoint, then what type of special segment is KD? Compare ADK and RDK. Do they have equal measures? Are they supplementary? What conclusion can you make? A R Compare your conjectures with the results of other students. Now combine the two conjectures from Steps 3 and 4 into one. Vertex Angle Bisector Conjecture In an isosceles triangle, the bisector of the vertex angle is also? and?. C-28 The properties you just discovered for isosceles triangles also apply to equilateral triangles. Equilateral triangles are also isosceles, although isosceles triangles are not necessarily equilateral. You have probably noticed the following property of equilateral triangles: When you construct an equilateral triangle, each angle measures 60°. If each angle measures 60°, then all three angles are congruent. So, if a triangle is equilateral, then it is equiangular. This is called the Equilateral Triangle Conjecture. If we agree that the Isosceles Triangle Conjecture is true, we can write the paragraph proof below. Paragraph Proof: The Equilateral Triangle Conjecture We need to show that if AB AC BC, then ABC is equiangular. By the Isosceles Triangle Conjecture, If AB AC, then mB mC. C, then B A If A C. B 242 CHAPTER 4 Discovering and Proving Triangle Properties If AB BC, then mA mC. C, then B A C If A. B If mA mC and mB mC, then mA mB mC. So, ABC is equiangular. C C and, then A B B C. B A If A The converse of the Equilateral Triangle Conjecture is called the Equiangular Triangle Conjecture, and it states: If a triangle is equiangular, then it is equilateral. Is this true? Yes, and the proof is almost identical to the proof above, except that you use the converse of the Isosce |
les Triangle Conjecture. So, if the Equilateral Triangle Conjecture and the Equiangular Triangle Conjecture are both true then we can combine them. Complete the conjecture below and add it to your conjecture list. Equilateral/Equiangular Triangle Conjecture Every equilateral triangle is?, and, conversely, every equiangular triangle is?. C-29 The Equilateral/Equiangular Triangle Conjecture is a biconditional conjecture: Both the statement and its converse are true. A triangle is equilateral if and only if it is equiangular. One condition cannot be true unless the other is also true. C A B EXERCISES In Exercises 1–3, ABC is isosceles with AC BC. You will need Construction tools for Exercise 10 1. Perimeter ABC 48 AC 18 AD? C 2. mABC 72° mADC? 3. mCAB 45° mACD? LESSON 4.8 Proving Isosceles Triangle Conjectures 243 In Exercises 4–6, copy the flowchart. Supply the missing statement and reasons in the proofs of Conjectures A, B, and C shown below. These three conjectures are all part of the Vertex Angle Bisector Conjecture. 4. Complete the flowchart proof for Conjecture A. Conjecture A: The bisector of the vertex angle in an isosceles triangle divides the isosceles triangle into two congruent triangles. C 1 2 Given: ABC is isosceles AC BC, and CD is the bisector of C Show: ADC BDC Flowchart Proof 1 CD is the bisector of C 2? Given Definition of angle bisector 3 ABC is isosceles with AC BC Given 4 CD CD Same segment A D B 5 ADC BDC? 5. Complete the flowchart proof for Conjecture B. C Conjecture B: The bisector of the vertex angle in an isosceles triangle is also the altitude to the base. Given: ABC is isosceles AC BC, and CD bisects C Show: CD is an altitude Flowchart Proof 1 ABC is isosceles with AC BC, and CD is the bisector of C Given 2 ADC BDC 4 1 2 Conjecture A? 3 1 and 2 form a linear pair 5 1 and 2 are supplementary Definition of linear pair Linear Pair Conjecture 1 2 D |
B A 6 1 and 2 are right angles Congruent supplementary angles are 90° 7 CD AB? 8? Definition of altitude 244 CHAPTER 4 Discovering and Proving Triangle Properties 6. Create a flowchart proof for Conjecture C. Conjecture C: The bisector of the vertex angle in an isosceles triangle is also the median to the base. Given: ABC is isosceles with AC BC CD is the bisector of C Show: CD is a median 7. In the figure at right, ABC, the plumb level is isosceles. A weight, called the plumb bob, hangs from a string attached at point C. If you place the level on a surface and the string is perpendicular to AB then the surface you are testing is level. To tell whether the string is perpendicular to AB, check whether it passes through the midpoint of AB. Create a flowchart proof to show that if D is the midpoint of AB, then CD is perpendicular to AB. Given: ABC is isosceles with AC BC D is the midpoint of AB A C D C B A B D Show: CD AB History Builders in ancient Egypt used a tool called a plumb level in building the great pyramids. With a plumb level, you can use the basic properties of isosceles triangles to determine whether a surface is level. 8. Write a paragraph proof of the Isosceles Triangle Conjecture. 9. Write a paragraph proof of the Equiangular Triangle Conjecture. 10. Construction Use compass and straightedge to construct a 30° angle. Review 11. Trace the figure below. Calculate the measure of each lettered angle. 3 4 1 2 and 3 n h g 4 70 52° 2 12. How many minutes after 3:00 will the hands of a clock overlap? LESSON 4.8 Proving Isosceles Triangle Conjectures 245 13. Find the equation of the line through point C that is parallel to side AB in ABC. The vertices are A(1, 3), B(4, 2), and C(6, 6). Write your answer in slope-intercept form, y mx b. 14. Sixty concurrent lines in a plane divide the plane into how many regions? 15. If two vertices of a triangle have coordinates A(1, 3) and B(7, 3), find the coordinates of point C so that ABC is a right triangle |
. Can you find any other points that would create a right triangle? 16. APPLICATION Hugo hears the sound of fireworks three seconds after he sees the flash. Duane hears the sound five seconds after he sees the flash. Hugo and Duane are 1.5 km apart. They know the flash was somewhere to the north. They also know that a flash can be seen almost instantly, but sound travels 340 m/sec. Do Hugo and Duane have enough information to locate the site of the fireworks? Make a sketch and label all the distances that they know or can calculate. 17. APPLICATION In an earlier exercise, you found the rule for the family of hydrocarbons called alkanes, or paraffins. These contain a straight chain of carbons. Alkanes can also form rings of carbon atoms. These molecules are called cycloparaffins. The first three cycloparaffins are shown below. Sketch. the molecule cycloheptane. Write the general rule for cycloparaffins CnH Cyclopropane Cyclobutane Cyclopentane H H H IMPROVING YOUR ALGEBRA SKILLS Number Tricks Try this number trick. Double the number of the month you were born. Subtract 16 from your answer. Multiply your result by 5, then add 100 to your answer. Subtract 20 from your result, then multiply by 10. Finally, add the day of the month you were born to your answer. The number you end up with shows the month and day you were born! For example, if you were born March 15th, your answer will be 315. If you were born December 7th, your answer will be 1207. Number tricks almost always involve algebra. Use algebra to explain why the trick works. 246 CHAPTER 4 Discovering and Proving Triangle Properties Napoleon’s Theorem In this exploration you’ll learn about a discovery attributed to French Emperor Napoleon Bonaparte (1769–1821). Napoleon was extremely interested in mathematics. This discovery, called Napoleon’s Theorem, uses equilateral triangles constructed on the sides of any triangle. Activity Napoleon Triangles Step 1 Open a new Sketchpad sketch. Draw ABC. B A C Step 2 Follow the Procedure Note to create a custom tool that constructs an equilateral triangle and its centroid given the endpoints of any segment. D B Step 3 Step 4 A C Use your custom tool on BC and CA. If an equilateral |
triangle falls inside your triangle, undo and try again, selecting the two endpoints in reverse order. Connect the centroids of the equilateral triangles. Triangle GQL is called the outer Napoleon triangle of ABC. Drag the vertices and the sides of ABC and observe what happens. Portrait of Napoleon by the French painter Anne-Louis Girodet (1767–1824) 1. Construct an equilateral triangle on AB. 2. Construct the centroid of the equilateral triangle. 3. Hide any medians or midpoints that you constructed for the centroid. 4. Select all three vertices, all three sides, and the centroid of the equilateral triangle. 5. Turn your new construction into a custom tool by choosing Create New Tool from the Custom Tools menu EXPLORATION Napoleon’s Theorem 247 Step 5 What can you say about the outer Napoleon triangle? Write what you think Napoleon discovered in his theorem. Here are some extensions to this theorem for you to explore. Step 6 Step 7 Construct segments connecting each vertex of your original triangle with the vertex of the equilateral triangle on the opposite side. What do you notice about these three segments? (This discovery was made by M. C. Escher.) Construct the inner Napoleon triangle by reflecting each centroid across its corresponding side in the original triangle. Measure the areas of the original triangle and of the outer and inner Napoleon triangles. How do these areas compare? LINES AND ISOSCELES TRIANGLES In this example, the lines y 3x 3 and y 3x 3 contain the sides of an isosceles triangle whose base is on the x-axis and whose line of symmetry is the y-axis. The window shown is {4.7, 4.7, 1, 3.1, 3.1, 1}. 1. Find other pairs of lines that form isosceles triangles whose bases are on the x-axis and whose lines of symmetry are the y-axis. 2. Find pairs of lines that form isosceles triangles whose bases are on the y-axis and whose lines of symmetry are the x-axis. 3. A line y mx b contains one side of an isosceles triangle whose base is on the x-axis and whose line of symmetry is the y-axis. What is the equation of the line containing the other side? Now suppose the line y mx b contains one side of an isosceles triangle whose base is on the |
y-axis and whose line of symmetry is the x-axis. What is the equation of the line containing the other side? 4. Graph the lines y 2x 2, y 1 x 1, y x, and y x. Describe the figure that 2 the lines form. Find other sets of lines that form figures like this one. 248 CHAPTER 4 Discovering and Proving Triangle Properties VIEW ● CHAPTER 11 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHA CHAPTER 4 R E V I E W In this chapter you made many conjectures about triangles. You discovered some basic properties of isosceles and equilateral triangles. You learned different ways to show that two triangles are congruent. Do you remember them all? Triangle congruence shortcuts are an important idea in geometry. You can use them to explain why your constructions work. In later chapters, you will use your triangle conjectures to investigate properties of other polygons. You also practiced reading and writing flowchart proofs. Can you sketch a diagram illustrating each conjecture you made in this chapter? Check your conjecture list to make sure it is up to date. Make sure you have a clear diagram illustrating each conjecture. EXERCISES 1. Why are triangles so useful in structures? 2. The first conjecture of this chapter is probably the most important so far. What is it? Why do you think it is so important? You will need Construction tools for Exercises 33, 34, and 37 3. What special properties do isosceles triangles have? 4. What does the statement “The shortest distance between two points is the straight line between them” have to do with the Triangle Inequality Conjecture? 5. What information do you need in order to determine that two triangles are congruent? That is, what are the four congruence shortcuts? 6. Explain why SSA is not a congruence shortcut. For Exercises 7–24, name the congruent triangles. State the conjecture or definition that supports the congruence statement. If you cannot show the triangles to be congruent from the information given, write “cannot be determined.” 7. PEA? P N E U TA 8. TOP? O Z T P A 9. MSE? E U S M O CHAPTER 4 REVIEW 249 EW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER |
4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTE 10. TIM? 11. TRP? 12. CAT? E I M T 13. CGH? H I G C N R T A P 14. AB CD ABE? B A E D C T C R A 15. Polygon CARBON is a regular hexagon. ACN? N O C B A R 16.??, AD? 17.?? 18 19.??, TR? 20.??, EI? 22.?? Is NCTM a parallelogram or a trapezoid? 23.?? LAI is isosceles with IA LA. M T N C A D R L 250 CHAPTER 4 Discovering and Proving Triangle Properties 21.?? Is WH a median? W O H Y 24.?? Is STOP a parallelogram? O Z P T I S ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 25. What’s wrong with this picture? 26. What’s wrong with this picture? N 38° 75° A G R G 6 cm O 7 cm W N 27. Quadrilateral CAMP has been divided into three triangles. Use the angle measures provided to determine the longest and shortest segments. 28. The measure of an angle formed by the bisectors of two angles in a triangle, as shown below, is 100°. What is angle measure x? C 120° a 30° g P 60° f e d M A 45° b c a a 100° b b x In Exercises 29 and 30, decide whether there is enough information to prove congruence. If there is, write a proof. If not, explain what is missing. 29. In the figure below, RE AE, S T, and ERL EAL. Is SA TR? S and MR FR. Is FRD isosceles? F 30. In the figure below, A M, AF FR 31. The measure of an angle formed by altitudes from two vertices of a triangle, as shown below, is 132°. What is angle measure x? 132° x 32. Connecting the legs of the chair at their midpoints as shown guarantees that the seat is parallel to the floor. Explain why. CHAPTER 4 REVIEW 251 EW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● |
CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTE For Exercises 33 and 34, use the segments and the angles below. Use either patty paper or a compass and a straightedge. The lowercase letter above each segment represents the length of the segment. x y z P A 33. Construction Construct PAL given P, A, and AL y. 34. Construction Construct two triangles PBS that are not congruent to each other given P, PB z, and SB x. 35. In the figure at right, is TI RE? Complete the flowchart proof or explain why they are not parallel. Given: M is the midpoint of both TE and IR. Show: TI RE Flowchart Proof??? 36. At the beginning of the chapter, you learned that triangles make structures more stable. Let’s revisit the shelves from Lesson 4.1. Explain how the SSS congruence shortcut guarantees that the shelves on the right will retain their shape, and why the shelves on the left wobble. 37. Construction Use patty paper or compass and straightedge to construct a 75° angle. Explain your method. 252 CHAPTER 4 Discovering and Proving Triangle Properties ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 TAKE ANOTHER LOOK 1. Explore the Triangle Sum Conjecture on a sphere or a globe. Can you draw a triangle that has two or more obtuse angles? Three right angles? Write an illustrated report of your findings. The section that follows, Take Another Look, gives you a chance to extend, communicate, and assess your understanding of the work you did in the investigations in this chapter. Sometimes it will lead to new, related discoveries. 2. Investigate the Isosceles Triangle Conjecture and the Equilateral⁄Equiangular Triangle Conjecture on a sphere. Write an illustrated report of your findings. 3. A friend claims that if the measure of one acute angle of a triangle is half the measure of another acute angle of the triangle, then the triangle can be divided into two isosceles triangles. Try this with a computer or other tools. Describe your method and explain why it works. 4. A friend claims that if one exterior angle has twice the measure of one of the remote interior angles, then the triangle is isosceles. Use a geometry software program or other tools to investigate this claim. |
Describe your findings. 5. Is there a conjecture (similar to the Triangle Exterior Angle Conjecture) that you can make about exterior and remote interior angles of a convex quadrilateral? Experiment. Write about your findings. 6. Is there a conjecture you can make about inequalities among the sums of the lengths of sides and/or diagonals of a quadrilateral? Experiment. Write about your findings. 7. In Chapter 3, you discovered how to construct the perpendicular bisector of a segment. Perform this construction. Now use what you’ve learned about congruence shortcuts to explain why this construction method works. 8. In Chapter 3, you discovered how to construct a perpendicular through a point on a line. Perform this construction. Use a congruence shortcut to explain why the construction works. 9. Is there a conjecture similar to the SSS Congruence Conjecture that you can make about congruence between quadrilaterals? For example, is SSSS a shortcut for quadrilateral congruence? Or, if three sides and a diagonal of one quadrilateral are congruent to the corresponding three sides and diagonal of another quadrilateral, must the two quadrilaterals be congruent (SSSD)? Investigate. Write a paragraph explaining how your conjectures follow from the triangle congruence conjectures you’ve learned. CHAPTER 4 REVIEW 253 EW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTE Assessing What You’ve Learned WRITE TEST ITEMS It’s one thing to be able to do a math problem. It’s another to be able to make one up. If you were writing a test for this chapter, what would it include? Start by having a group discussion to identify the key ideas in each lesson of the chapter. Then divide the lessons among group members, and have each group member write a problem for each lesson assigned to them. Try to create a mix of problems in your group, from simple one-step exercises that require you to recall facts to more complex, multistep problems that require more thinking. An example of a simple problem might be finding a missing angle measure in a triangle. A more complex problem could be a flowchart for a logical argument, or a word problem that requires using geometry to model a real-world situation. Share your problems with your group members and try out one |
another’s problems. Then discuss the problems in your group: Were they representative of the content of the chapter? Were some too hard or too easy? Writing your own problems is an excellent way to assess and review what you’ve learned. Maybe you can even persuade your teacher to use one of your items on a real test! ORGANIZE YOUR NOTEBOOK Review your notebook to be sure it is complete and well organized. Write a one-page chapter summary based on your notes. WRITE IN YOUR JOURNAL Write a paragraph or two about something you did in this class that gave you a great sense of accomplishment. What did you learn from it? What about the work makes you proud? UPDATE YOUR PORTFOLIO Choose a piece of work from this chapter to add to your portfolio. Document the work, explaining what it is and why you chose it. PERFORMANCE ASSESSMENT While a classmate, a friend, a family member, or your teacher observes, perform an investigation from this chapter. Explain each step, including how you arrived at the conjecture. 254 CHAPTER 4 Discovering and Proving Triangle Properties CHAPTER 5 Discovering and Proving Polygon Properties The mathematicians may well nod their heads in a friendly and interested manner—I still am a tinkerer to them. And the “artistic” ones are primarily irritated. Still, maybe I’m on the right track if I experience more joy from my own little images than from the most beautiful camera in the world...” Still Life and Street, M. C. Escher, 1967–1968 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● study properties of polygons ● discover relationships among their angles, sides, and diagonals ● learn about real-world applications of special polygons L E S S O N 5.1 I find that the harder I work, the more luck I seem to have. THOMAS JEFFERSON Polygon Sum Conjecture There are many kinds of triangles, but in Chapter 4, you discovered that the sum of their angle measures is always 180°. In this lesson you’ll investigate the sum of the angle measures in quadrilaterals, pentagons, and other polygons. Then you’ll look for a pattern in the sum of the angle measures in any polygon. Investigation Is There a Polygon Sum Formula? For this investigation each person in |
your group should draw a different version of the same polygon. For example, if your group is investigating hexagons, try to think of different ways you could draw a hexagon. Step 1 Step 2 Step 3 Draw the polygon. Carefully measure all the interior angles, then find the sum. Share your results with your group. If you measured carefully, you should all have the same sum! If your answers aren’t exactly the same, find the average. Copy the table below. Repeat Steps 1 and 2 with different polygons, or share results with other groups. Complete the table. Number of sides of polygon 3 4 5 6 7 8 … n Sum of measures of angles 180° … You can now make some conjectures. Quadrilateral Sum Conjecture The sum of the measures of the four angles of any quadrilateral is?. Pentagon Sum Conjecture The sum of the measures of the five angles of any pentagon is?. C-30 C-31 256 CHAPTER 5 Discovering and Proving Polygon Properties If a polygon has n sides, it is called an n-gon. Step 4 Look for a pattern in the completed table. Write a general formula for the sum of the angle measures of a polygon in terms of the number of sides, n. Polygon Sum Conjecture The sum of the measures of the n interior angles of an n-gon is?. C-32 You used inductive reasoning to discover the formula. Now you can use deductive reasoning to see why the formula works. Step 5 Draw all the diagonals from one vertex of your polygon. How many triangles do the diagonals create? How does the number of triangles relate to the formula you found? How can you check that your formula is correct for a polygon with 12 sides? D d e qQ A a Step 6 Write a short paragraph proof of the Quadrilateral Sum Conjecture. Use the diagram of quadrilateral QUAD. (Hint: Use the Triangle Sum Conjecture.) vu U EXERCISES 1. Use the Polygon Sum Conjecture to complete the table. You will need Geometry software for Exercise 19 Number of sides of polygon 7 8 9 10 11 20 55 100 Sum of measures of angles 2. What is the measure of each angle of an equiangular pentagon? An equiangular hexagon? Complete the table. 5 6 7 8 9 10 12 16 100 Number of |
sides of equiangular polygon Measures of each angle of equiangular polygon In Exercises 3–8, use your conjectures to calculate the measure of each lettered angle. 3. a? 4. b? 76° 72° a 70° b 110° 116° 68° 5. e? f? e f LESSON 5.1 Polygon Sum Conjecture 257 6. c? d? 7. g? h? 8. j? k? d c 44° 78° 30° 66° 130° 108° h 117° g k 38° j 9. What’s wrong with this picture? 10. What’s wrong with this picture? 11. Three regular polygons meet at point A. How many sides does the largest polygon have? 82° 102° 76° 135° 154° 49° A 12. Trace the figure at right. Calculate each lettered angle measure. 13. How many sides does a polygon have if the sum of its angle measures is 2700°? 14. How many sides does an equiangular polygon have if each interior angle measures 156°? p e 36° 1 2 d 98° 60° 106° m n 116° a b c 116° g f 122° 138° h 77° 1 2 k j 87° 15. Archaeologist Ertha Diggs has uncovered a piece of a ceramic plate. She measures it and finds that each side has the same length and each angle has the same measure. She conjectures that the original plate was the shape of a regular polygon. She knows that if the original plate was a regular 16-gon, it was probably a ceremonial dish from the third century. If it was a regular 18-gon, it was probably a palace dinner plate from the twelfth century. If each angle measures 160°, from what century did the plate likely originate? 258 CHAPTER 5 Discovering and Proving Polygon Properties 16. APPLICATION You need to build a window frame for an octagonal window like this one. To make the frame, you’ll cut identical trapezoidal pieces. What are the measures of the angles of the trapezoids? Explain how you found these measures. 17. Use this diagram to prove the Pentagon Sum Conjecture Review 18. This figure is a detail of one vertex of the tiling at the beginning of this lesson. Find the missing angle measure x. x 60° 19. Technology Use geometry software to construct |
a quadrilateral and locate the midpoints of its four sides. Construct segments connecting the midpoints of opposite sides. Construct the point of intersection of the two segments. Drag a vertex or a side so that the quadrilateral becomes concave. Observe these segments and make a conjecture. 20. Write the equation of the perpendicular bisector of the segment with endpoints (12, 15) and (4, 3). 21. ABC has vertices A(0, 0), B(4, 2), and C(8, 8). What is the equation of the median to side AB? P 22. Line is parallel to AB. As P moves to the right along, which of these measures will always increase? A. The distance PA B. The measure of APB C. The perimeter of ABP D. The measure of ABP A B IMPROVING YOUR VISUAL THINKING SKILLS Net Puzzle The clear cube shown has the letters DOT printed on one face. When a light is shined on that face, the image of DOT appears on the opposite face. The image of DOT on the opposite face is then painted. Copy the net of the cube and sketch the painted image of the word, DOT, on the correct square and in the correct position. DOT DOT LESSON 5.1 Polygon Sum Conjecture 259 Exterior Angles of a Polygon In Lesson 5.1, you discovered a formula for the sum of the measures of the interior angles of any polygon. In this lesson you will discover a formula for the sum of the measures of the exterior angles of a polygon. Set of exterior angles L E S S O N 5.2 If someone had told me I would be Pope someday, I would have studied harder. POPE JOHN PAUL I Best known for her participation in the Dada Movement, German artist Hannah Hoch (1889–1978) painted Emerging Order in the Cubist style. Do you see any examples of exterior angles in the painting? You will need ● a straightedge ● a protractor Investigation Is There an Exterior Angle Sum? Let’s use some inductive and deductive reasoning to find the exterior angle measures in a polygon. Each person in your group should draw the same kind of polygon for Steps 1–5. Step 1 Step 2 Step 3 Step 4 Draw a large polygon. Extend its sides to form a set of exterior angles. Measure all the interior angles of the polygon except one. |
Use the Polygon Sum Conjecture to calculate the measure of the remaining interior angle. Check your answer using your protractor. Use the Linear Pair Conjecture to calculate the measure of each exterior angle. Calculate the sum of the measures of the exterior angles. Share your results with your group members. 260 CHAPTER 5 Discovering and Proving Polygon Properties Step 5 Repeat Steps 1–4 with different kinds of polygons, or share results with other groups. Make a table to keep track of the number of sides and the sum of the exterior angle measures for each kind of polygon. Find a formula for the sum of the measures of a polygon’s exterior angles. Exterior Angle Sum Conjecture For any polygon, the sum of the measures of a set of exterior angles is?. C-33 Step 6 Step 7 Step 8 Study the software construction above. Explain how it demonstrates the Exterior Angle Sum Conjecture. Using the Polygon Sum Conjecture, write a formula for the measure of each interior angle in an equiangular polygon. Using the Exterior Angle Sum Conjecture, write the formula for the measure of each exterior angle in an equiangular polygon. Step 9 Using your results from Step 8, you can write the formula for an interior angle a different way. How do you find the measure of an interior angle if you know the measure of its exterior angle? Complete the next conjecture. Equiangular Polygon Conjecture C-34 You can find the measure of each interior angle of an equiangular n-gon by using either of these formulas:? or?. LESSON 5.2 Exterior Angles of a Polygon 261 EXERCISES 1. Complete this flowchart proof of the Exterior Angle Sum Conjecture for a triangle. Flowchart Proof 1 a b 180°? 2 c d 180 = °? Addition property of equality 3 e f 180° 5 a + c + e = °??? = °? Subtraction property of equality 2. What is the sum of the measures of the exterior angles of a decagon? 3. What is the measure of an exterior angle of an equiangular pentagon? An equiangular hexagon? In Exercises 4–9, use your new conjectures to calculate the measure of each lettered angle. 4. a 7. 140° 60° 68° 84° g f e h 5. 8. 68° 43° |
b 44° c b 56° a d 94° 69° 6. 9. d c 86° g 39° d f c e a k 18° h b 10. How many sides does a regular polygon have if each exterior angle measures 24°? 11. How many sides does a polygon have if the sum of its interior angle measures is 7380°? 12. Is there a maximum number of obtuse exterior angles that any polygon can have? If so, what is the maximum? If not, why not? Is there a minimum number of acute interior angles that any polygon must have? If so, what is the minimum? If not, why not? 262 CHAPTER 5 Discovering and Proving Polygon Properties Technology The aperture of a camera is an opening shaped like a regular polygon surrounded by thin sheets that form a set of exterior angles. These sheets move together or apart to close or open the aperture, limiting the amount of light passing through the camera’s lens. How does the sequence of closing apertures shown below demonstrate the Exterior Angle Sum Conjecture? Does the number of sides make a difference in the opening and closing of the aperture? Review 13. Name the regular polygons that appear in the tiling below. Find the measures of the angles that surround point A in the tiling. 14. Name the regular polygons that appear in the tiling below. Find the measures of the angles that surround any vertex point in the tiling. A 15. RAC DCA, CD AR, AC DR. Is AD CR? Why? D R A C 16. DT RT, DA RA. Is D R? Why? D R A T IMPROVING YOUR VISUAL THINKING SKILLS Dissecting a Hexagon II Make six copies of the hexagon at right by tracing it onto your paper. Then divide each hexagon into twelve identical parts in a different way. LESSON 5.2 Exterior Angles of a Polygon 263 Star Polygons If you arrange a set of points roughly around a circle or an oval, and then you connect each point to the next with segments, you should get a convex polygon like the one at right. What do you get if you connect every second point with segments? You get a star polygon like the ones shown in the activity below. In this activity, you’ll investigate the angle measure sums of star polygons. Activity Exploring Star Polyg |
ons -pointed star ABCDE 6-pointed star FGHIJK Draw five points A through E in a circular path, clockwise. Connect every second point with AC, CE, EB, BD, and DA. Measure the five angles A through E at the star points. Use the calculator to find the sum of the angle measures. Drag each vertex of the star and observe what happens to the angle measures and the calculated sum. Does the sum change? What is the sum? Copy the table on page 265. Use the Polygon Sum Conjecture to complete the first column. Then enter the angle sum for the 5-pointed star. Step 1 Step 2 Step 3 Step 4 Step 5 264 CHAPTER 5 Discovering and Proving Polygon Properties Step 6 Step 7 Step 8 Repeat Steps 1–5 for a 6-pointed star. Enter the angle sum in the table. Complete the column for each n-pointed star with every second point connected. What happens if you connect every third point to form a star? What would be the sum of the angle measures in this star? Complete the table column for every third point. Use what you have learned to complete the table. What patterns do you notice? Write the rules for n-pointed stars. Angle measure sums by how the star points are connected Number of star points Every point Every 2nd point Every 3rd point Every 4th point Every 5th point 5 6 7 540° 720° Step 9 Step 10 Let’s explore Step 4 a little further. Can you drag the vertices of each star polygon to make it convex? Describe the steps for turning each one into a convex polygon, and then back into a star polygon again, in the fewest steps possible. In Step 9, how did the sum of the angle measure change when a polygon became convex? When did it change? This blanket by Teresa ArchuletaSagel is titled My Blue Vallero Heaven. Are these star polygons? Why? EXPLORATION Star Polygons 265 L E S S O N 5.3 Imagination is the highest kite we fly. LAUREN BACALL Kite and Trapezoid Properties Recall that a kite is a quadrilateral with exactly two distinct pairs of congruent consecutive sides. If you construct two different isosceles triangles on opposite sides of a common base and then remove the base, you have constructed a kite. In an isos |
celes triangle, the vertex angle is the angle between the two congruent sides. Therefore, let’s call the two angles between each pair of congruent sides of a kite the vertex angles of the kite. Let’s call the other pair the nonvertex angles. Nonvertex angles Vertex angles You will need ● patty paper Step 1 Step 2 A kite also has one line of reflectional symmetry, just like an isosceles triangle. You can use this property to discover other properties of kites. Let’s investigate. Investigation 1 What Are Some Properties of Kites? In this investigation you will look at angles and diagonals in a kite to see what special properties they have. On patty paper, draw two connected segments of different lengths, as shown. Fold through the endpoints and trace the two segments on the back of the patty paper. Compare the size of each pair of opposite angles in your kite by folding an angle onto the opposite angle. Are the vertex angles congruent? Are the nonvertex angles congruent? Share your observations with others near you and complete the conjecture. Step 1 Step 2 266 CHAPTER 5 Discovering and Proving Polygon Properties Kite Angles Conjecture The? angles of a kite are?. Step 3 Draw the diagonals. How are the diagonals related? Share your observations with others in your group and complete the conjecture. Kite Diagonals Conjecture The diagonals of a kite are?. C-35 C-36 What else seems to be true about the diagonals of kites? Step 4 Compare the lengths of the segments on both diagonals. Does either diagonal bisect the other? Share your observations with others near you. Copy and complete the conjecture. Kite Diagonal Bisector Conjecture The diagonal connecting the vertex angles of a kite is the? of the other diagonal. Step 5 Fold along both diagonals. Does either diagonal bisect any angles? Share your observations with others and complete the conjecture. Kite Angle Bisector Conjecture The? angles of a kite are? by a?. C-37 C-38 Pair of base angles You will prove the Kite Diagonal Bisector Conjecture and the Kite Angle Bisector Conjecture as exercises after this lesson. Bases Let’s move on to trapezoids. Recall that a trapez |
oid is a quadrilateral with exactly one pair of parallel sides. In a trapezoid the parallel sides are called bases. A pair of angles that share a base as a common side are called base angles. Pair of base angles In the next investigation, you will discover some properties of trapezoids. LESSON 5.3 Kite and Trapezoid Properties 267 Science A trapezium is a quadrilateral with no two sides parallel. The words trapezoid and trapezium come from the Greek word trapeza, meaning table. There are bones in your wrists that anatomists call trapezoid and trapezium because of their geometric shapes. Trapezium Investigation 2 What Are Some Properties of Trapezoids? You will need ● a straightedge ● a protractor ● a compass This is a view inside a deflating hot-air balloon. Notice the trapezoidal panels that make up the balloon. Step 1 Step 2 Use the two edges of your straightedge to draw parallel segments of unequal length. Draw two nonparallel sides connecting them to make a trapezoid. Use your protractor to find the sum of the measures of each pair of consecutive angles between the parallel bases. What do you notice about this sum? Share your observations with your group. Find sum. Step 3 Copy and complete the conjecture. Trapezoid Consecutive Angles Conjecture The consecutive angles between the bases of a trapezoid are?. C-39 Recall from Chapter 3 that a trapezoid whose two nonparallel sides are the same length is called an isosceles trapezoid. Next, you will discover a few properties of isosceles trapezoids. 268 CHAPTER 5 Discovering and Proving Polygon Properties Step 4 Step 5 Like kites, isosceles trapezoids have one line of reflectional symmetry. Through what points does the line of symmetry pass? Use both edges of your straightedge to draw parallel lines. Using your compass, construct two congruent segments. Connect the four segments to make an isosceles trapezoid. Measure each pair of base angles. What do you notice about the pair of base angles in each trapezoid? Compare your observations with others near you. Compare. Compare. Step 6 Copy and complete the conjecture. Isosceles Trapezoid Conjecture The base angles of an isosceles trapezoid are?. What other parts of an isosceles trapez |
oid are congruent? Let’s continue. Step 7 Draw both diagonals. Compare their lengths. Share your observations with others near you. Step 8 Copy and complete the conjecture. Isosceles Trapezoid Diagonals Conjecture The diagonals of an isosceles trapezoid are?. C-40 C-41 Suppose you assume that the Isosceles Trapezoid Conjecture is true. What pair of triangles and which triangle congruence conjecture would you use to explain why the Isosceles Trapezoid Diagonals Conjecture is true? EXERCISES Use your new conjectures to find the missing measures. 1. Perimeter? 12 cm 20 cm 2. x? y? x 146° y 47° You will need Construction tools for Exercises 10–12 3. x? y? 128° y x LESSON 5.3 Kite and Trapezoid Properties 269 4. x? Perimeter 85 cm x 37 cm 18 cm 5. x? y? 18° y x 29° 6. x? y? Perimeter 164 cm y 12 cm x y 12 cm y 81° 7. Sketch and label kite KITE with vertex angles K and T and KI TE. Which angles are congruent? 8. Sketch and label trapezoid QUIZ with one base QU. What is the other base? Name the two pairs of base angles. 9. Sketch and label isosceles trapezoid SHOW with one base SH. What is the other base? Name the two pairs of base angles. Name the two sides of equal length. In Exercises 10–12, use the properties of kites and trapezoids to construct each figure. You may use either patty paper or a compass and a straightedge. 10. Construction Construct kite BENF given sides BE and EN and diagonal BN. How many different kites are possible? B B E E N N 11. Construction Given W, I, base WI, and nonparallel side IS, construct trapezoid WISH. I W S I I W 12. Construction Construct a trapezoid BONE with BO NE. How many different trapezoids can you construct? B N O E 13. Write a paragraph proof or flowchart proof showing how the Kite Diagonal Bisector Conjecture logically follows from the Converse of the Perpendicular Bisector Conjecture. 270 CH |
graphing calculator. This is done with parametric equations, which give the x- and y-coordinates of a point in terms of a third variable, or parameter, t. Set your calculator’s mode to degrees and parametric. Set a friendly window with an x-range of 4.7 to 4.7 and a y-range of 3.1 to 3.1. Set a t-range of 0 to 360, and t-step of 60. Enter the equations x 3 cos t and y 3 sin t, and graph them. You should get a hexagon. The equations you graphed are actually the parametric equations for a circle. By using a t-step of 60 for t-values from 0 to 360, you tell the calculator to compute only six points for the circle. Use your calculator to investigate the following. Summarize your findings. Choose different t-steps to draw different regular polygons, such as an equilateral triangle, a square, a regular pentagon, and so on. What is the measure of each central angle of an n-gon? What happens as the measure of each central angle of a regular polygon decreases? What happens as you draw polygons with more and more sides? Experiment with rotating your polygons by choosing different t-min and t-max values. For example, set a t-range of 45 to 315, then draw a square. Find a way to draw star polygons on your calculator. Can you explain how this works? 272 CHAPTER 5 Discovering and Proving Polygon Properties L E S S O N 5.4 Research is formalized curiosity. It is poking and prying with a purpose. ZORA NEALE HURSTON Properties of Midsegments As you learned in Chapter 3, the segment connecting the midpoints of two sides of a triangle is the midsegment of a triangle. The segment connecting the midpoints of the two nonparallel sides of a trapezoid is also called the midsegment of a trapezoid. In this lesson you will discover special properties of midsegments. Investigation 1 Triangle Midsegment Properties You will need ● patty paper In this investigation you will discover two properties of the midsegment of a triangle. Each person in your group can investigate a different triangle. Step 1 Step 2 Step 3 Step 1 Step 2 Step 3 Draw a triangle on a piece of patty paper. Pinch the patty paper to locate midpoints of the sides. |
Draw the midsegments. You should now have four small triangles. Place a second piece of patty paper over the first and copy one of the four triangles. Compare all four triangles by sliding the copy of one small triangle over the other three triangles. Compare your results with the results of your group. Copy and complete the conjecture. Three Midsegments Conjecture The three midsegments of a triangle divide it into?. C-42 Step 4 Mark all the congruent angles in your drawing. What conclusions can you make about each midsegment and the large triangle’s third side, using the Corresponding Angles Conjecture and the Alternate Interior Angles Conjecture? What do the other students in your group think? LESSON 5.4 Properties of Midsegments 273 Step 5 Compare the length of the midsegment to the large triangle’s third side. How do they relate? Copy and complete the conjecture. Triangle Midsegment Conjecture A midsegment of a triangle is? to the third side and? the length of?. C-43 In the next investigation, you will discover two properties of the midsegment of a trapezoid. Investigation 2 Trapezoid Midsegment Properties Each person in your group can investigate a different trapezoid. Make sure you draw the two bases perfectly parallel. You will need ● patty paper 3 4 1 2 3 4 1 2 Step 1 Step 2 Step 3 Draw a small trapezoid on the left side of a piece of patty paper. Pinch the paper to locate the midpoints of the nonparallel sides. Draw the midsegment. Label the angles as shown. Place a second piece of patty paper over the first and copy the trapezoid and its midsegment. Compare the trapezoid’s base angles with the corresponding angles at the midsegment by sliding the copy up over the original. Are the corresponding angles congruent? What can you conclude about the midsegment and the bases? Compare your results with the results of other students. The midsegment of a triangle is half the length of the third side. How does the length of the midsegment of a trapezoid compare to the lengths of the two bases? Let’s investigate. On the original trapezoid, extend the longer base to the right by at least the length of the shorter base. Slide the second patty paper under the first. Show the sum of the |
lengths of the two bases by marking a point on the extension of the longer base. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 274 CHAPTER 5 Discovering and Proving Polygon Properties Sum Step 5 Step 6 Step 7 Step 7 How many times does the midsegment fit onto the segment representing the sum of the lengths of the two bases? What do you notice about the length of the midsegment and the sum of the lengths of the two bases? Step 8 Combine your conclusions from Steps 4 and 7 and complete this conjecture. Trapezoid Midsegment Conjecture The midsegment of a trapezoid is? to the bases and is equal in length to?. C-44 What happens if one base of the trapezoid shrinks to a point? Then the trapezoid collapses into a triangle, the midsegment of the trapezoid becomes a midsegment of the triangle, and the Trapezoid Midsegment Conjecture becomes the Triangle Midsegment Conjecture. Do both of your midsegment conjectures work for the last figure EXERCISES 1. How many midsegments does a triangle have? A trapezoid have? 2. What is the perimeter of TOP? T 3. x? y? You will need Construction tools for Exercises 9 and 18 4. z? 65° P 8 R O 20 10 A x 60° y 40° z 42° LESSON 5.4 Properties of Midsegments 275 7. q? 5. What is the perimeter of TEN. m? n? p? 36 cm n 73° p m 51° 48 cm 8. Copy and complete the flowchart to show that LN RD. F N Given: Midsegment LN in FOA Midsegment RD in IOA Show: LN RD Flowchart Proof 1 2 FOA with midsegment LN Given IOA with midsegment RD Given 3 LN OA? 4? Triangle Midsegment Conjecture L O D R I 5? Two lines parallel to the same line are parallel 13 24 q A 9. Construction When you connected the midpoints of the three sides of a triangle in Investigation 1, you created four congruent triangles. Draw a quadrilateral on patty paper and pinch the paper to locate the midpoints of the four sides. Connect the midpoints to form a quadrilateral. What special type of quadrilateral do you get when you connect the |
midpoints? Use the Triangle Midsegment Conjecture to explain your answer. 10. Deep in a tropical rain forest, archaeologist Ertha Diggs and her assistant researchers have uncovered a square-based truncated pyramid (a square pyramid with the top part removed). The four lateral faces are isosceles trapezoids. A line of darker mortar runs along the midsegment of each lateral face. Ertha and her co-workers make some measurements and find that one of these midsegments measures 41 meters and each bottom base measures 52 meters. Now that they have this information, Ertha and her team can calculate the length of the top base without having to climb up and measure it. Can you? What is the length of the top edge? How do you know? 276 CHAPTER 5 Discovering and Proving Polygon Properties 11. Ladie and Casey pride themselves on their estimation skills and take turns estimating distances. Casey claims that two large redwood trees visible from where they are sitting are 180 feet apart, and Ladie says they are 275 feet apart. The problem is, they can’t measure the distance to see whose estimate is better, because their cabin is located between the trees. All of a sudden, Ladie recalls her geometry: “Oh yeah, the Triangle Midsegment Conjecture!” She collects a tape measure, a hammer, and some wooden stakes. What is she going to do? b i n C a Review 12. The 40-by-60-by-80 cm sealed rectangular container shown at right is resting on its largest face. It is filled with a liquid to a height of 30 cm. Sketch the container resting on its smallest face. Show the height of the liquid in this new position. 13. Write the converse of this statement: If exactly one 40 cm 30 cm 60 cm 80 cm diagonal bisects a pair of opposite angles of a quadrilateral, then the quadrilateral is a kite. Is the converse true? Is the original statement true? If either conjecture is not true, sketch a counterexample. 14. Trace the figure below. Calculate the measure of each lettered angle. 1 2 a b 54 15. CART is an isosceles trapezoid. What are 16. HRSE is a kite. What are the coordinates the coordinates of point T? of point R? y T (?,?) R (12, 8) y E (0, 8) H |
(5, 0) S (10, 0) x C (0, 0) x A (15, 0) R (?,?) LESSON 5.4 Properties of Midsegments 277 17. Find the coordinates of midpoints E and Z. Show that the slope of the line containing midsegment EZ is equal to the slope of the line containing YT. 18. Construction Use the kite properties you discovered in Lesson 5.3 to construct kite FRNK given diagonals RK and FN and side NK. Is there only one solution? y R (4, 7) Z (?,?) E (?,?) Y (0, 0) T (8, 3) x R F N N K K BUILDING AN ARCH In this project, you’ll design and build your own Roman arch. Arches can have a simple semicircular shape, or a pointed “broken arch” shape. Horseshoe Arch Basket Arch Tudor Arch Lancet Arch In arch construction, a wooden support holds the voussoirs in place until the keystone is placed (see arch diagram on page 271). It’s said that when the Romans made an arch, they would make the architect stand under it while the wooden support was removed. That was one way to be sure architects carefully designed arches that wouldn’t fall! What size arch would you like to build? Decide the dimensions of the opening, the thickness of the arch, and the number of voussoirs. Decide on the materials you will use. You should have your trapezoid and your materials approved by your group or your teacher before you begin construction. Your project should include A scale diagram that shows the exact size and angle of the voussoirs and the keystone. A template for your voussoirs. Your arch. The arches in this Roman aqueduct, above the Gard River in France, are typical of arches you can find throughout regions that were once part of the Roman Empire. An arch can carry a lot of weight, yet it also provides an opening. The abutments on the sides of the arch keep the arch from spreading out and falling down. b c a b b c 278 CHAPTER 5 Discovering and Proving Polygon Properties L E S S O N 5.5 If there is an opinion, facts will be found to support it. JUDY SPROLES Properties of Parallelograms In this |
lesson you will discover some special properties of parallelograms. A parallelogram is a quadrilateral whose opposite sides are parallel. Rhombuses, rectangles, and squares all fit this definition as well. Therefore, any properties you discover for parallelograms will also apply to these other shapes. However, to be sure that your conjectures will apply to any parallelogram, you should investigate parallelograms that don’t have any other special properties, such as right angles, all congruent angles, or all congruent sides. You will need ● graph paper ● patty paper or a compass ● a straightedge ● a protractor Investigation Four Parallelogram Properties First you’ll create a parallelogram. E V L O Step 1 Step 2 Step 1 Using the lines on a piece of graph paper as a guide, draw a pair of parallel lines that are at least 6 cm apart. Using the parallel edges of your straightedge, make a parallelogram. Label your parallelogram LOVE. Step 2 Let’s look at the opposite angles. Measure the angles of parallelogram LOVE. Compare a pair of opposite angles using patty paper or your protractor. Compare results with your group. Copy and complete the conjecture. Parallelogram Opposite Angles Conjecture The opposite angles of a parallelogram are?. C-45 Two angles that share a common side in a polygon are consecutive angles. In parallelogram LOVE, LOV and EVO are a pair of consecutive angles. The consecutive angles of a parallelogram are also related. Step 3 Find the sum of the measures of each pair of consecutive angles in parallelogram LOVE. LESSON 5.5 Properties of Parallelograms 279 Share your observations with your group. Copy and complete the conjecture. Parallelogram Consecutive Angles Conjecture The consecutive angles of a parallelogram are?. C-46 Step 4 Step 5 Describe how to use the two conjectures you just made to find all the angles of a parallelogram with only one angle measure given. Next let’s look at the opposite sides of a parallelogram. With your compass or patty paper, compare the lengths of the opposite sides of the parallelogram you made. Share your results with your group. Copy and complete the conjecture. Parallelogram Opposite Sides Conjecture The opposite sides of a parallelogram are?. C-47 |
Step 6 Step 7 Finally, let’s consider the diagonals of a parallelogram. Construct the diagonals LV and EO, as shown below. Label the point where the two diagonals intersect point M. Measure LM and VM. What can you conclude about point M? Is this conclusion also true for diagonal EO? How do the diagonals relate? E V M Share your results with your group. Copy and complete the conjecture. L O Parallelogram Diagonals Conjecture The diagonals of a parallelogram?. C-48 Parallelograms are used in vector diagrams, which have many applications in science. A vector is a quantity that has both magnitude and direction. Vectors describe quantities in physics, such as velocity, acceleration, and force. You can represent a vector by drawing an arrow. The length and direction of the arrow represent the magnitude and direction of the vector. For example, a velocity vector tells you an airplane’s speed and direction. The lengths of vectors in a diagram are proportional to the quantities they represent. Engine velocity 560 mi/hr Wind velocity 80 mi/hr 280 CHAPTER 5 Discovering and Proving Polygon Properties In many physics problems, you combine vector quantities acting on the same object. For example, the wind current and engine thrust vectors determine the velocity of an airplane. The resultant vector of these vectors is a single vector that has the same effect. It can also be called a vector sum. To find a resultant vector, make a parallelogram with the vectors as sides. The resultant vector is the diagonal of the parallelogram from the two vectors’ tails to the opposite vertex. The resultant vector represents the actual speed and direction of the plane. Ve Vector represents engine velocity. Ve Vw Vw Vector represents wind velocity. In the diagram at right, the resultant vector shows that the wind will speed up the plane, and will also blow it slightly off course. EXERCISES Use your new conjectures in the following exercises. In Exercises 1–6, each figure is a parallelogram. 1. c? d? 27 cm 34 cm d c 4. VF 36 m EF 24 m EI 42 m What is the perimeter of NVI? E V N 2. a? b? b 48° a 5. What is the perimeter? x 3 x 3 17 F I 7. Construction Given side LA, side AS, and L, construct parallelog |
. What is the measure of each angle in the isosceles trapezoid face of a voussoir in this 15-stone arch? a? b? LESSON 5.5 Properties of Parallelograms 283 19. Is XYW WYZ? Explain. 20. Sketch the section formed when this pyramid is sliced by the plane. 58° W 83° 58° X Z 83° 39° 39° Y 21. Technology Construct two segments that bisect each other. Connect their endpoints. What type of quadrilateral is this? Draw a diagram and explain why. 22. Technology Construct two intersecting circles. Connect the two centers and the two points of intersection to form a quadrilateral. What type of quadrilateral is this? Draw a diagram and explain why. IMPROVING YOUR VISUAL THINKING SKILLS A Puzzle Quilt Fourth-grade students at Public School 95, the Bronx, New York, made the puzzle quilt at right with the help of artist Paula Nadelstern. Each square has a twin made of exactly the same shaped pieces. Only the colors, chosen from traditional Amish colors, are different. For example, square A1 is the twin of square B3. Match each square with its twin. A B C D 1 2 3 4 284 CHAPTER 5 Discovering and Proving Polygon Properties ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 5 ● USING YO USING YOUR ALGEBRA SKILLS 5 Solving Systems of Linear Equations A system of equations is a set of two or more equations with the same variables. The solution of a system is the set of values that makes all the equations in the system true. For example, the system of equations below has solution (2, 3). Verify this by substituting 2 for x and 3 for y in both equations. y 2x 7 y 3x 3 Graphically, the solution of a system is the point of intersection of the graphs of the equations. You can estimate the solution of a system by graphing the equations. However, the point of intersection may not have convenient integer coordinates. To find the exact solution, you can use algebra. Examples A and B review how to use the substitution and elimination methods for solving systems of equations. X = 2 Y = –3 Use the substitution method to solve the system 3y 12x 21. |
12x 2y 1 Start by solving the first equation for y to get y 4x 7. Now, substitute the expression 4x 7 from the resulting equation for y in the second original equation. 12x 2y 1 12x 2(4x – 7) 1 x 3 4 Substitute 4x 7 for y. Second original equation. Solve for x. EXAMPLE A Solution To find y, substitute 3 for x in either original equation. 4 21 3y 123 4 y 4 Solve for y. Substitute 3 for x in the first original equation. 4 The solution of the system is 3, 4. Verify by substituting these values for x 4 and y in each of the original equations. EXAMPLE B The band sold calendars to raise money for new uniforms. Aisha sold 6 desk calendars and 10 wall calendars for a total of $100. Ted sold 12 desk calendars and 4 wall calendars for a total of $88. Find the price of each type of calendar by writing a system of equations and solving it using the elimination method. USING YOUR ALGEBRA SKILLS 5 Solving Systems of Linear Equations 285 ALGEBRA SKILLS 5 ● USING YOUR ALGEBRA SKILLS 5 ● USING YOUR ALGEBRA SKILLS 5 ● USING YO Solution Let d be the price of a desk calendar, and let w be the price of a wall calendar. You can write this system to represent the situation. 6d 10w 100 12d 4w 88 Aisha’s sales. Ted’s sales. Solving a system by elimination involves adding or subtracting the equations to eliminate one of the variables. To solve this system, first multiply both sides of the first equation by 2. 6d 10w 100 12d 4w 88 12d 20w 200 12d 4w 88 → Now, subtract the second equation from the first to eliminate d. 12d 20w 200 (12d 4w 88) 16w 112 w 7 To find the value of d, substitute 7 for w in either original equation. The solution is w 7 and d 5, so a wall calendar costs $7 and a desk calendar costs $5. EXERCISES Solve each system of equations algebraically. 2. 1. 4. 3. y 2x 2 6x 2y 3 5x y 1 15x 2y x 2y 3 2x y 16 4x 3y 3 7x 9y |
6 For Exercises 5 and 6 solve the systems. What happens? Graph each set of equations and use the graphs to explain your results. 5. x 6y 10 1 x 3y 5 2 6. 2x y 30 y 2x 1 7. A snowboard rental company offers two different rental plans. Plan A offers $4/hr for the rental and a $20 lift ticket. Plan B offers $7/hr for the rental and a free lift ticket. a. Write the two equations that represent the costs for the two plans, using x for the number of hours. Solve for x and y. b. Graph the two equations. What does the point of intersection represent? c. Which is the better plan if you intend to snowboard for 5 hours? What is the most number of hours of snowboarding you can get for $50? x, and y 4 x, y 1 8. The lines y 3 2 x 3 intersect to form a triangle. 3 3 3 Find the vertices of the triangle. 286 CHAPTER 5 Discovering and Proving Polygon Properties L E S S O N 5.6 You must know a great deal about a subject to know how little is known about it. LEO ROSTEN Properties of Special Parallelograms The legs of the lifting platforms shown at right form rhombuses. Can you visualize how this lift would work differently if the legs formed parallelograms that weren’t rhombuses? In this lesson you will discover some properties of rhombuses, rectangles, and squares. What you discover about the diagonals of these special parallelograms will help you understand why these lifts work the way they do. Investigation 1 What Can You Draw with the Double-Edged Straightedge? In this investigation you will discover the special parallelogram that you can draw using just the parallel edges of a straightedge. You will need ● patty paper ● a double-edged straightedge Step 1 Step 2 Step 3 Step 1 Step 2 Step 3 On a piece of patty paper, use a double-edged straightedge to draw two pairs of parallel lines that intersect each other. Assuming that the two edges of your straightedge are parallel, you have drawn a parallelogram. Place a second patty paper over the first and copy one of the sides of the parallelogram. Compare the length of the side on the second patty paper with the lengths of the other three sides of the parallelogram. How do they compare |
? Share your results with your group. Copy and complete the conjecture. Double-Edged Straightedge Conjecture C-49 If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a?. LESSON 5.6 Properties of Special Parallelograms 287 In Chapter 3, you learned how to construct a rhombus using a compass and straightedge, or using patty paper. Now you know a quicker and easier way, using a double-edged straightedge. To construct a parallelogram that is not a rhombus, you need two double-edged staightedges of different widths. Rhombus Parallelogram Now let’s investigate some properties of rhombuses. Investigation 2 Do Rhombus Diagonals Have Special Properties? You will need ● patty paper Step 1 Step 2 Step 1 Step 2 Draw in both diagonals of the rhombus you created in Investigation 1. Place the corner of a second patty paper onto one of the angles formed by the intersection of the two diagonals. Are the diagonals perpendicular? Compare your results with your group. Also, recall that a rhombus is a parallelogram and that the diagonals of a parallelogram bisect each other. Combine these two ideas into your next conjecture. Rhombus Diagonals Conjecture The diagonals of a rhombus are?, and they?. Step 3 The diagonals and the sides of the rhombus form two angles at each vertex. Fold your patty paper to compare each pair of angles. What do you observe? Compare your results with your group. Copy and complete the conjecture. Rhombus Angles Conjecture The? of a rhombus? the angles of the rhombus. C-50 C-51 288 CHAPTER 5 Discovering and Proving Polygon Properties So far you’ve made conjectures about a quadrilateral with four congruent sides. Now let’s look at quadrilaterals with four congruent angles. What special properties do they have? Recall the definition you created for a rectangle. A rectangle is an equiangular parallelogram. Here is a thought experiment. What is the measure of each angle of a rectangle? The Quadrilateral Sum Conjecture says all four angles add up to 360°. They� |
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