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Chapter 1: Tools of Geometry

Chapter 1: Tools of Geometry. What geometric terms are you familiar with?. Lesson 1-1 Patterns and Inductive Reasoning. Check Skills You’ll Need. (For help, go the Skills Handbook, page 753.). Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . . Some are even and some are odd.

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Chapter 1: Tools of Geometry

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  1. Chapter 1: Tools of Geometry What geometric terms are you familiar with?

  2. Lesson 1-1 Patterns and Inductive Reasoning Check Skills You’ll Need (For help, go the Skills Handbook, page 753.) Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . . Some are even and some are odd. 1. Make a list of the positive even numbers.  2. Make a list of the positive odd numbers.  3. Copy and extend this list to show the first 10 perfect squares. 12 = 1, 22 = 4, 32 = 9, 42 = 16, . . . 4. Which do you think describes the square of any odd number? It is odd. It is even.

  3. Lesson 1-1 Patterns and Inductive Reasoning

  4. Lesson 1-1 Patterns and Inductive Reasoning Key Concepts Inductive reasoningis A conjectureis A counterexample to a conjecture is

  5. Lesson 1-1 Patterns and Inductive Reasoning Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence. Example 1 384, 192, 96, 48, …

  6. Lesson 1-1 Patterns and Inductive Reasoning Make a conjecture about the sum of the cubes of the first 25 counting numbers. Example 2 Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern. 13 = 1 = 12 = 12 13 + 23 = 9 = 32 = (1 + 2)2 13 + 23 + 33 = 36 = 62 = (1 + 2 + 3)2 13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2 13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2

  7. Lesson 1-1 Patterns and Inductive Reasoning Example 3 Find a counterexample for each conjecture. a. A number is always greater than its reciprocal. b. If a number is divisible by 5, then it is divisible by 10.

  8. Lesson 1-1 Patterns and Inductive Reasoning The price of overnight shipping was $8.00 in 2000, $9.50 in 2001, and $11.00 in 2002. Make a conjecture about the price in 2003. Example 4

  9. Lesson 1-1 Patterns and Inductive Reasoning Lesson Quiz Find a pattern for each sequence. Use the pattern to show the next two terms or figures. 1. 3, –6, 18, –72, 360 2.

  10. Lesson 1-1 Patterns and Inductive Reasoning Lesson Quiz (continued) Use the table and inductive reasoning. 3. Find the sum of the first 10 counting numbers. 4. Find the sum of the first 1000 counting numbers. Show that the conjecture is false by finding one counterexample. 5. The sum of two prime numbers is an even number.

  11. Lesson 1-1 Patterns and Inductive Reasoning Homework Pages 6-8 19-28, 32-52 even

  12. Lesson 1-2 Drawing, Nets, and Other Models Check Skills You’ll Need (For help, go to Lesson 1-1.) Draw the next figure in each sequence. 1. 2.

  13. Lesson 1-2 Drawing, Nets, and Other Models

  14. Lesson 1-2 Drawing, Nets, and Other Models Key Concepts Anisometric drawingof a three-dimensional object An orthographic drawing is A foundation drawingshows A netis

  15. Lesson 1-2 Drawing, Nets, and Other Models Example 1 Make an isometric drawing of the cube structure below.

  16. Lesson 1-2 Drawing, Nets, and Other Models Make an orthographic drawing of the isometric drawing below. Example 2

  17. Lesson 1-2 Drawing, Nets, and Other Models Example 3 Create a foundation drawing for the isometric drawing below.

  18. Lesson 1-2 Drawing, Nets, and Other Models Is the pattern a net for a cube? If so, name two letters that will be on opposite faces. Example 4

  19. Lesson 1-2 Drawing, Nets, and Other Models Draw a net for the figure with a square base and four isosceles triangle faces. Label the net with its dimensions. Example 5

  20. Lesson 1-2 Drawing, Nets, and Other Models Lesson Quiz Use the figure at the right for Exercises 1–2. 1. Make an isometric drawing of the cube structure. 2. Make an orthographic drawing.

  21. Lesson 1-2 Drawing, Nets, and Other Models Lesson Quiz (continued) 3. Is the pattern a net for a cube? If so, name two letters that will be on opposite faces. 4. Draw a net for the figure.

  22. Lesson 1-2 Drawing, Nets, and Other Models Homework Pages 13-14 1-16, 18-20, 23-26

  23. Lesson 1-3Points, Lines, and Planes Check Skills You’ll Need (For help, go to the Skills Handbook, page 760.) 1.y = x + 5 2.y = 2x – 4  3.y = 2x y = –x + 7 y = 4x – 10 y = –x + 15 4. Copy the diagram of the four points A, B, C, and D. Draw as many different lines as you can to connect pairs of points. Solve each system of equations.

  24. Lesson 1-3Points, Lines, and Planes

  25. Lesson 1-3Points, Lines, and Planes Key Concepts • Three basic undefined terms: • point • line • plane Point • A point is a location or position. • A point has no size. • It is represented by a small dot and is named by a capital letter. • A geometric figure is a set of points. • Space is defined as the set of all points.

  26. t Lesson 1-3Points, Lines, and Planes Key Concepts Line • A lineis • You can name a line by any two points on the line or by a single lowercase script letter • Points that lie on the same line are SG or GS or line t

  27. P Plane P Plane ABC Lesson 1-3Points, Lines, and Planes Key Concepts Plane A planeis • You can name a plane by a single capital letter. • Planes can also be named by at least three of its noncollinear points. • Points and lines in the same plane are

  28. In the figure below, name three points that are collinear and three points that are not collinear. Lesson 1-3Points, Lines, and Planes Example 1

  29. Lesson 1-3Points, Lines, and Planes Example 2 Name the plane shown in two different ways.

  30. B F G C E A D H Lesson 1-3Points, Lines, and Planes Example 3 a) Name two different planes that contain points C and G. b) Name all the planes that contain point E.

  31. t Lesson 1-3Points, Lines, and Planes Key Concepts A postulate or axiomis Postulate: Through any two points there is exactly one line. Line t is the only line that passes through points A and B

  32. C Lesson 1-3Points, Lines, and Planes Key Concepts Postulate: If two lines intersect, then they intersect in exactly one point. Lines AE and BD intersect at C

  33. M N Lesson 1-3Points, Lines, and Planes Key Concepts Postulate:If two planes intersect, then they intersect in exactly one line. Plane M and plane N intersect in RS

  34. Lesson 1-3Points, Lines, and Planes Key Concepts Postulate:Through any three noncollinear points there is exactly one plane.

  35. Use the diagram below. What is the intersection of plane HGC and plane AED? Lesson 1-3Points, Lines, and Planes Example 4

  36. Lesson 1-3Points, Lines, and Planes Lesson Quiz Use the diagram at right. 1. Name three collinear points. 2. Name two different planes that contain points C and G. 3. Name the intersection of plane AED and plane HEG. 4. How many planes contain the points A, F, and H? 5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line.

  37. Lesson 1-3Points, Lines, and Planes Homework Pages 19 – 21; 1 – 24, 30 – 52 even, 55 – 60

  38. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Check Skills You’ll Need (For help, go to Lesson 1-3.) 1.2.3. 4. the bottom 5. the top 6. the front 7. the back 8. the left side 9. the right side Judging by appearances, will the lines intersect? Name the plane represented by each surface   of the box.

  39. Lesson 1-4Segments, Rays, Parallel Lines, and Planes

  40. Segment AB A B Lesson 1-4Segments, Rays, Parallel Lines, and Planes Key Concepts A line segmentis

  41. Ray YX Y Lesson 1-4Segments, Rays, Parallel Lines, and Planes Key Concepts A rayis

  42. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Key Concepts Opposite raysare RQ and RS are opposite rays.

  43. Name the segments and rays in the figure. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Example 1

  44. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Key Concepts Parallel linesare Skew lines

  45. Use the figure below. Name all segments that are parallel to AE. Name all segments that are skew to AE. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Example 2

  46. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Identify a pair of parallel planes in your classroom. Example 3

  47. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Lesson Quiz Use the figure below for Exercises 1-3. 1. Name the segments that form the triangle. 2. Name the rays that have point T as their endpoint. 3. Explain how you can tell that no lines in the figure are parallel or skew.

  48. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Lesson Quiz (continued) Use the figure below for Exercises 4 and 5. 4. Name a pair of parallel planes. 5. Name a line that is skew to XW.

  49. Lesson 1-4Segments, Rays, Parallel Lines, and Planes Homework Pages 25-27 1-35, 39, 41-45 Quiz 1-1 through 1-4 Friday, Sept. 9

  50. Lesson 1-5Measuring Segments Check Skills You’ll Need (For help, go to the Skills Handbook, pages 757 and 758.) Simplify each absolute value expression. 1. |–6| 2. |3.5| 3. |7 – 10| 4. |–4 – 2| 5. |–2 – (–4)| 6. |–3 + 12| 7.x + 2x – 6 = 6 8. 3x + 9 + 5x = 81 9.w – 2 = –4 + 7w Solve each equation.

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