25. How would you make the octahedral die shown below?
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1 304450_ch_08_enqxd 12/6/06 1:39 PM Page 577 Chapter Summary 577 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 304450_ch_08_enqxd 12/6/06 1:39 PM Page CHAPTER 8 / Geometry as Shape 6. 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 and applications drawings, representations of buildings, and nets. BASIC CONCEPTS Section 8.1 Basic Concepts of Geometry hand-eye coordination 505 figure-ground perception 505 perceptual constancy 505 visual discrimination 506 visual memory 506 Point, line, plane, on, and between are undefined terms. 508 Sets of points may be collinear or noncollinear. 509 Sets of points may be coplanar or not; lines may be coplanar or not. 510 line 510 line segment 510 ray 510 endpoint 510 intersect 511 Relationships between lines: perpendicular, parallel, concurrent, skew 498 Angles have vertices and sides. 512 Angles have an interior and an exterior. 512 Naming angles with one letter, one number, three letters 513 Measuring angles protractor 513 degree 513 Kinds of angles right 516 acute 516 obtuse 516 straight 516 reflex 516 adjacent 516 complementary 516 supplementary 516 vertical 516 Section 8.2 Two-Dimensional Figures Kinds of lines and curves closed curves 527 simple curves 526 Any simple closed curve partitions the plane into 3 disjoint regions. 528 polygon 529 sides 529 vertex (vertices) 529 base 529 congruent 529 Classifying triangles By length of sides: equilateral, isosceles, scalene 531 By size of angle: right triangle, obtuse triangle, acute triangle 531 Special line segments in triangles angle bisector 535 median 535 altitude 536 perpendicular bisector 535 Congruence congruent 537 Quadrilaterals trapezoid 539 parallelogram 539 kite 539 rhombus 540 rectangle 540 square 540 Polygons convex or concave 545 regular polygon 546 diagonal 540 interior angle 547 exterior angle 548 central angle 548 Properties The sum of the measures of the angles of a convex polygon 180 n Circles circle 549 radius (radii) 550 diameter 550 chord 550 tangent 550 arc 550 Section 8.3 Three-Dimensional Figures space figure 563 polyhedron 563 polyhedra 563 solids 563 Parts face 563 edge 563 vertex 563
3 304450_ch_08_enqxd 12/6/06 1:39 PM Page 579 Chapter 8 Review Exercises 579 Classifying convex 564 concave 564 Polyhedra prism 565 box 565 base 565 cube 565 lateral face 565 dihedral 565 right prism 565 oblique prism 565 Pyramid pyramid 566 apex 566 Regular polyhedra tetrahedron 567 icosahedron 567 octahedron 567 dodecahedron 567 Relationships among polyhedron Euler s formula 569 Connecting two-dimensional representations to threedimensional objects different views 569 isometric drawings 569 cross sections 569 nets 569 Cylinders, Cones, Spheres cylinder 574 apex 574 oblique cylinder 574 right cylinder 574 right cone 574 oblique cone 574 sphere 575 center 575 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 304450_ch_08_enqxd 12/6/06 1:40 PM Page CHAPTER 8 / Geometry as Shape 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 304450_ch_08_enqxd 12/6/06 1:40 PM Page 581 Chapter 8 Review Exercises 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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