25. How would you make the octahedral die shown below?
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1 304150_ch_08_enqxd 1/16/04 6:11 AM Page CHAPTER 8 / Geometry as Shape draw others you will not necessarily need all of them. Describe your method, other than random trial and error. How confident are you that you have drawn all of them? Why? 24. Which of the nets below is a possible net for an oatmeal box (which has the shape of a cylinder)? 21. Look at the triangular prism shown below. If you were to make a net for this polyhedron, what would be the dimensions of the ramp? Look at the figure at the right below. If you were to make a net for this polyhedron, what would be the exact dimensions of the roof? 25. How would you make the octahedral die shown below? 1" 2" 1" 2" 6" 4" 22. Below are drawings of two polyhedr Draw the top, front, and side views for each polyhedron. 26. Most doors are rectangles. Occasionally, however, you see a door whose shape is a nonrectangular polygon. In Lord of the Rings, the door to Frodo s house was roun Explain why we don t see round doors very often! 23. Which of the nets below is a possible net for a cereal box? CHAPTER SUMMARY 1. Important spatial thinking abilities include: eyemotor coordination, figure-ground perception, perceptual constancy, visual discrimination, and visual memory. 2. The van Hiele levels of geometric thinking help us to see that the level of our geometric thinking determines what we see (for example, the churn dash investigation) and how powerfully geometric ideas can be use 3. Our current knowledge of geometry took thousands of years to develop and is the result of intuitive thinking, inductive thinking, and deductive thinking. The Greeks were the first people to develop mathematical systems, that is, a coherent system of mathematical ideas. 4. Our understanding of geometry has been built carefully on a foundation of axioms, undefined terms, definitions, and theorems. The building blocks of this knowledge are the ideas about points, lines, planes, and space. 5. Shapes are found everywhere, and virtually all shapes have functional and/or aesthetic value.
2 304150_ch_08_enqxd 1/16/04 6:11 AM Page 565 Chapter Summary Each geometric shape represents the common characteristics of a set of objects. For example, the set of objects we call prisms can look very different to a novice, but they are all prisms because they have two bases that are polygons, parallel, and congruent, and they have faces that are parallelograms. 7. Every shape has multiple attributes which means that we can classify any set of shapes in multiple ways. Recognizing and understanding these attributes helps us to understand the shape more deeply and leads to practical applications. 8. Classifying leads us to deeper understanding of mathematical structure which leads to greater mathematical power, for example, that the sum of the angles of any polygon is equal to 180 n Looking for and recognizing relationships within and between shapes also leads to understanding of mathematical structure; for example, the many relationships among quadrilaterals (which we will further develop in Chapter 9), and seeing the relationships between prisms and cylinders. 10. Coordinate geometry is a useful tool for understanding attributes of shapes and relationships among shapes. 11. There are rich connections between twodimensional and three-dimensional shapes. This knowledge has many practical uses applications drawings, representations of buildings, and nets. BASIC CONCEPTS Section 8.1 Basic Concepts of Geometry hand-eye coordination 492 figure-ground perception 492 perceptual constancy 492 visual discrimination 493 visual memory 493 Point, line, plane, on, and between are undefined terms. 494 Sets of points may be collinear or noncollinear. 496 Sets of points may be coplanar or not; lines may be coplanar or not. 497 line 497 line segment 497 ray 497 endpoint 497 intersect 498 Relationships between lines: perpendicular, parallel, concurrent, skew 498 Angles have vertices and sides. 499 Angles have an interior and an exterior. 499 Naming angles with one letter, one number, three letters 500 Measuring angles protractor 500 degree 500 Kinds of angles right 503 acute 503 obtuse 503 straight 503 reflex 503 adjacent 503 complementary 503 supplementary 503 vertical 503 Section 8.2 Two-Dimensional Figures Kinds of lines and curves closed curves 514 simple curves 513 Any simple closed curve partitions the plane into 3 disjoint regions. 515 polygon 516 sides 516 vertex (vertices) 516 base 520 congruent 524 Classifying triangles By length of sides: equilateral, isosceles, scalene 518 By size of angle: right triangle, obtuse triangle, acute triangle 518 Special line segments in triangles angle bisector 522 median 522 altitude 523 perpendicular bisector 522 Congruence congruent 524 Quadrilaterals trapezoid 526 parallelogram 526 kite 526 rhombus 527 rectangle 527 square 527 Polygons convex or concave 532 regular polygon 533 diagonal 532 interior angle 534 exterior angle 535 central angle 535 Properties The sum of the measures of the angles of a convex polygon 180 n Circles circle 536 radius (radii) 537 diameter 537 chord 537 tangent 537 arc 537 Section 8.3 Three-Dimensional Figures space figure 550 polyhedron 550 polyhedra 550 solids 550 Parts face 550 edge 550 vertex 550
3 304150_ch_08_enqxd 1/16/04 6:11 AM Page CHAPTER 8 / Geometry as Shape Classifying convex 551 concave 551 Polyhedra prism 552 box 552 base 552 cube 552 lateral face 552 dihedral 552 right prism 552 oblique prism 552 Pyramid pyramid 553 apex 553 Regular polyhedra tetrahedron 554 icosahedron 554 octahedron 554 dodecahedron 554 Relationships among polyhedron Euler s formula 555 Connecting two-dimensional representations to threedimensional objects different views 556 isometric drawings 556 cross sections 556 nets 556 Cylinders, Cones, Spheres cylinder 561 apex 561 oblique cylinder 561 right cylinder 561 right cone 561 oblique cone 561 sphere 562 center 562 CHAPTER 8 REVIEW EXERCISES 1. Without using a protractor, draw an angle that is approximately 30 degrees. Explain your reasoning. Now measure the angle. Repeat the process for a 120-degree angle. i 2. In the figure at the right, AB and BC i A are perpendicular G lines. C B Name two complementary angles. F D E Name two supplementary angles. Name two vertical angles. Name two adjacent angles. 3. True or false? If true, briefly explain why. If false, provide a counterexample. If three distinct lines intersect, then they are coplanar. If two lines do not intersect, then they are parallel. 4. Why is it necessary to start with undefined terms in geometry? 5. A teacher defined triangle as a shape made by three line segments. By that definition, the shape at the right is a triangle. Fix the definition so that it works. 6. How many different quadrilaterals can you make that will fit on a 3 3 Geoboard? 7. How could you convince someone that the sum of the angles of any quadrilateral is 360 degrees? 8. What attributes do all rectangles have in common? 9. For each of the following, draw the figure or explain why it is impossible. A triangle that is isosceles and obtuse.
4 304150_ch_08_enqxd 1/16/04 6:11 AM Page 567 Chapter 8 Review Exercises 567 A quadrilateral that has exactly one set of parallel sides and two right angles. A nonregular hexagon that has all sides congruent. A pentagon with three right angles and two sets of parallel sides. e. A concave pentagon. 10. How many right angles can a hexagon have? 11. Write down all the attributes of each of the following figures. Then write down the attributes they have in common. 12. Name six different polygons that you can see in this figure in such a way that the reader can easily find the polygons that you have foun 13. The four figures at the left below are Kiwis. The four figures in the middle group are not Kiwis. Kiwis Not Kiwis Kiwis or Not Kiwis Judging on the basis of the four Kiwis and the four not-kiwis, list the attributes that all Kiwis possess. Which of the four figures in the group at the right are Kiwis? Justify your answers. 14. Write directions for making the figure below. Following your directions, the reader should be able to make the same figure. 15. Is the Venn diagram below a valid representation of the relationship between parallelograms and rectangles? If yes, explain why. If not, explain why not. Parallelograms Rectangles 16. Draw a Venn diagram to represent the relationship between regular polygons and convex polygons. 17. Find the distance between the following pairs of points: 2, 4 and 4, Find the midpoint of the line segments connecting these pairs of points: 0, 3 and 5, Three vertices of a square are 5, 0, 5, 6 and 8, 3. What are the coordinates of the fourth vertex? 20. I am a parallelogram. Two of my sides are parallel to the bottom of the paper. The vertices of my bases are at 2, 3 and 8, 3. The slope of the line segments that form my sides is 0.75, and the length of each side is 5 units. What are my other two vertices? 21. Identify the number of vertices, edges, and faces of the accompanying figure.
5 304150_ch_08_enqxd 1/16/04 6:11 AM Page CHAPTER 8 / Geometry as Shape 22. What attributes do all cones and pyramids have in common that not all three-dimensional figures have? 23. Write directions for making the block building at the right any way you want. Write directions using a different metho 24. Sketch the figure at the right on Isometric Dot Paper. 25. Sketch the front, side, and top view of the building above. 26. Following are the top view, the front view, and the right-side view of a building created only by cubes. Without making the building, predict how many cubes it will have. Explain your prediction. Predict the left-hand view. Explain your prediction. Sketch the building on Isometric Dot Paper. Front Right-side Top 27. Draw a net for a prism whose base is a right isosceles triangle. Draw a different net for the same prism. 28. Write a definition of diagonal for polyhedr 29. Describe the polygon formed by the following cross sections. You cut a cross section that is perpendicular to the base of a square pyramid but does not pass through the apex of the pyrami You cut a cross section that is perpendicular to the base of a circular cylinder. 30. Below is a drawing of a polyhedron. Draw the top, front, and side views for this polyhedron.
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