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## Patterns and Inductive Reasoning

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**Patterns and Inductive Reasoning**GEOMETRY LESSON 1-1 (For help, go the Skills Handbook, page 715.) 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. 1-1**Patterns and Inductive Reasoning**GEOMETRY LESSON 1-1 Solutions 1. Even numbers end in 0, 2, 4, 6, or 8: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . . 2. Odd numbers end in 1, 3, 5, 7, or 9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, . . . 3. 12 = (1)(1) = 1; 22 = (2)(2) = 4; 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; 62 = (6)(6) = 36; 72 = (7)(7) = 49; 82 = (8)(8) = 64; 92 = (9)(9) = 81; 102 = (10)(10) = 100 4. The odd squares in Exercise 3 are all odd, so the square of any odd number is odd. 1-1**Find a pattern for the sequence. Use the pattern to show**the next two terms in the sequence. 384, 192, 96, 48, … Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Each term is half the preceding term. So the next two terms are 48 ÷ 2 = 24 and 24 ÷ 2 = 12. 1-1**Make a conjecture about the sum of the cubes of the first 25**counting numbers. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 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 The sum of the first two cubes equals the square of the sum of the first two counting numbers. 1-1**(continued)**The sum of the first three cubes equals the square of the sum of the first three counting numbers. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 This pattern continues for the fourth and fifth rows of the table. 13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2 13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2 So a conjecture might be that the sum of the cubes of the first 25 counting numbers equals the square of the sum of the first 25 counting numbers, or (1 + 2 + 3 + … + 25)2. 1-1**The first three odd prime numbers are 3, 5, and 7. Make and**test a conjecture about the fourth odd prime number. Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 One pattern of the sequence is that each term equals the preceding term plus 2. So a possible conjecture is that the fourth prime number is 7 + 2 = 9. However, because 3 X 3 = 9 and 9 is not a prime number, this conjecture is false. The fourth prime number is 11. 1-1**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. Write the data in a table. Find a pattern. 2000 2001 2002 $8.00 $9.50 $11.00 Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 Each year the price increased by $1.50. A possible conjecture is that the price in 2003 will increase by $1.50. If so, the price in 2003 would be $11.00 + $1.50 = $12.50. 1-1**Pages 6–9 Exercises**1. 80, 160 2. 33,333; 333,333 3. –3, 4 4. , 5. 3, 0 6. 1, 7. N, T 8. J, J 9. 720, 5040 10. 64, 128 11. , 12. , 13. James, John 14. Elizabeth, Louisa 15. Andrew, Ulysses 16. Gemini, Cancer 17. 18. 1 5 1 6 19. The sum of the first 6 pos. even numbers is 6 • 7, or 42. 20. The sum of the first 30 pos. even numbers is 30 • 31, or 930. 21. The sum of the first 100 pos. even numbers is 100 • 101, or 10,100. 1 16 1 32 1 3 1 36 1 49 Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 1-1**Patterns and Inductive Reasoning**GEOMETRY LESSON 1-1 31. 31, 43 32. 10, 13 33. 0.0001, 0.00001 34. 201, 202 35. 63, 127 36. , 37. J, S 38. CA, CO 39. B, C 28. ÷ = and is improper. 29. 75°F 30. 40 push-ups; answers may vary. Sample: Not very confident, Dino may reach a limit to the number of push-ups he can do in his allotted time for exercises. 1 2 1 3 3 2 3 2 22. The sum of the first 100 odd numbers is 1002, or 10,000. 23. 555,555,555 24. 123,454,321 25–28.Answers may vary. Samples are given. 25. 8 + (–5 = 3) and 3 > 8 26. • > and • > 27. –6 – (–4) < –6 and –6 – (–4) < –4 31 32 63 64 / 1 3 1 3 1 2 1 2 1 3 1 2 / / 1-1**40. Answers may vary.**Sample: In Exercise 31, each number increases by increasing multiples of 2. In Exercise 33, to get the next term, divide by 10. 41. You would get a third line between and parallel to the first two lines. 47. Answers may vary. Samples are given. a. Women may soon outrun men in running competitions. b. The conclusion was based on continuing the trend shown in past records. c. The conclusions are based on fairly recent records for women, and those rates of improvement may not continue. The conclusion about the marathon is most suspect because records date only from 1955. 42. 43. 44. 45. 46. 102 cm Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 1-1**52. 21, 34, 55**53. a. Leap years are years that are divisible by 4. b. 2020, 2100, and 2400 c. Leap years are years divisible by 4, except the final year of a century which must be divisible by 400. So, 2100 will not be a leap year, but 2400 will be. 48. a. b. about 12,000 radio stations in 2010 c. Answers may vary. Sample: Confident; the pattern has held for several decades. 49. Answers may vary. Sample: 1, 3, 9, 27, 81, . . . 1, 3, 5, 7, 9, . . . Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 50. His conjecture is probably false because most people’s growth slows by 18 until they stop growing somewhere between 18 and 22 years. 51.a. b. H and I c. a circle 1-1**54. Answers may vary.**Sample: 100 + 99 + 98 + … + 3 + 2 + 1 1 + 2 + 3 + … + 98 + 99 + 100 101 + 101 + 101 + … + 101 + 101 + 101 The sum of the first 100 numbers is , or 5050. The sum of the first n numbers is . 55.a. 1, 3, 6, 10, 15, 21 b. They are the same. c. The diagram shows the product of n and n + 1 divided by 2 when n = 3. The result is 6. 55. (continued) d. 56. B 57. I 58.[2]a. 25, 36, 49 b.n2 [1] one part correct 100 • 101 2 n(n+1) 2 Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 1-1**59.[4]a. The product of 11 and a three-digit number that**begins and ends in 1 is a four-digit number that begins and ends in 1 and has middle digits that are each one greater than the middle digit of the three-digit number. (151)(11) = 1661 (161)(11) = 1771 b. 1991 c. No; (191)(11) = 2101 59. (continued) [3] minor error in explanation [2] incorrect description in part (a) [1] correct products for (151)(11), (161)(11), and (181)(11) 60-67. 68.B 69.N 70.G Patterns and Inductive Reasoning GEOMETRY LESSON 1-1 1-1**Patterns and Inductive Reasoning**GEOMETRY LESSON 1-1 Find a pattern for each sequence. Use the pattern to show the next two terms or figures. 1. 3, –6, 18, –72, 360 2. 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. –2160; 15,120 55 500,500 Sample: 2+3=5, and 5 is not even 1-1**Points, Lines, and Planes**GEOMETRY LESSON 1-2 (For help, go to the Skills Handbook, page 722.) 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. 1-2**Solutions**1. By substitution, x + 5 = –x + 7; adding x – 5 to both sides results in 2x = 2; dividing both sides by 2 results in x = 1. Since x = 1, y = (1) + 5 = 6. (x, y) = (1, 6) 2. By substitution, 2x – 4 = 4x – 10; adding –4x + 4 to both sides results in –2x = –6; dividing both sides by –2 results in x = 3. Since x = 3, y = 2(3) – 4 = 6 – 4 = 2. (x, y) = (3, 2) 3. By substitution, 2x = –x + 15; adding x to both sides results in 3x = 15; dividing both sides by 3 results in x = 5. Since x = 5, y = 2(5) = 10. (x, y) = (5, 10) 4. The 6 different lines are AB, AC, AD, BC, BD, and CD. Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**In the figure below, name three points that are collinear**and three points that are not collinear. Points Y, Z, and W lie on a line, so they are collinear. Points, Lines, and Planes GEOMETRY LESSON 1-2 Any other set of three points do not lie on a line, so no other set of three points is collinear. For example, X, Y, and Z and X, W, and Z form triangles and are not collinear. 1-2**Name the plane shown in two different ways.**Points, Lines, and Planes GEOMETRY LESSON 1-2 You can name a plane using any three or more points on that plane that are not collinear. Some possible names for the plane shown are the following: plane RST plane RSU plane RTU plane STU plane RSTU 1-2**Use the diagram below. What is the intersection of plane HGC**and plane AED? The back and left faces of the cube intersect at HD. Planes HGC and AED intersect vertically at HD. Points, Lines, and Planes GEOMETRY LESSON 1-2 As you look at the cube, the front face is on plane AEFB, the back face is on plane HGC, and the left face is on plane AED. 1-2**Shade the plane that**contains X, Y, and Z. Points X, Y, and Z are the vertices of one of the four triangular faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X, Y, and Z. Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**9. Answers may vary.**Sample: AE, EC, GA 10. Answers may vary. Sample: BF, CD, DF 11.ABCD 12.EFHG 13.ABHF 14. EDCG 15.EFAD 16.BCGH 17.RS 18.VW 19.UV 20.XT 21. planes QUX and QUV 22. planes XTS and QTS 23. planes UXT and WXT 24.UVW and RVW Points, Lines, and Planes GEOMETRY LESSON 1-2 Pages 13–16 Exercises 1. no 2. yes; line n 3. yes; line n 4. yes; line m 5. yes; line n 6. no 7. no 8. yes; line m 1-2**25.**26. 27. 28. 29. 30. S 31.X 32.R 33.Q 34. X Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**35. no**36. yes 37. no 38. coplanar 39. coplanar 40. noncoplanar 41. coplanar 42. noncoplanar 43. noncoplanar 44. Answers may vary. Sample: The plane of the ceiling and the plane of a wall intersect in a line. 45. Through any three noncollinear points there is exactly one plane. The ends of the legs of the tripod represent three noncollinear points, so they rest in one plane. Therefore, the tripod won’t wobble. 46. Postulate 1-1: Through any two points there is exactly one line. 47. Answer may vary. Sample: 48. 49. not possible Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**56.**no 57. no 58. yes 50. 51. not possible 52. yes 53. yes 54. no 55. yes Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**59.**yes 60. always 61. never 62. always 63. always 64. sometimes 65. never 66. a. 1 b. 1 c. 1 d. 1 e. A line and a point not on the line are always coplanar. 67. Post. 1-4: Through three noncollinear points there is exactly one plane. 68. Answers may vary. Sample: Post. 1-3: If two planes intersect, then they intersect in exactly one line. 69. A, B, and D 70. Post. 1-1: Through any two points there is exactly one line. Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**71. Post. 1-3: If two planes intersect, then they intersect**in exactly one line. 72. The end of one leg might not be coplanar with the ends of the other three legs. (Post. 1-4) 73. yes 74. yes 75. no 76. no 77. yes Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**78.**no 79. Infinitely many; explanations may vary. Sample: Infinitely many planes can intersect in one line. 80. By Post. 1-1, points D and B determine a line and points A and D determine a line. The distress signal is on both lines and, by Post. 1-2, there can be only one distress signal. 81.a. Since the plane is flat, the line would have to curve so as to contain the 2 points and not lie in the plane; but lines are straight. b. One plane; Points A, B, and C are noncollinear. By Post. 1-4, they are coplanar. Then, by part (a), AB and BC are coplanar. 82. 1 Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**1**4 83. 84. 1 85.A 86.I 87. B 88.H 89.[2] a.ABD, ABC, ACD, BCD b.AD, BD, CD [1] one part correct 90. The pattern 3, 9, 7, 1 repeats 11 times for n = 1 to 44. For n = 45, the last digit is 3. 91. I, K 92. 42, 56 93. 1024, 4096 94. 25, –5 95. 34 96. 44 Points, Lines, and Planes GEOMETRY LESSON 1-2 1-2**HE**Points, Lines, and Planes GEOMETRY LESSON 1-2 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. D, J, and H planes BCGF and CGHD 1 Sample: Planes AEHD and BFGC never intersect. 1-2**Segments, Rays, Parallel Lines and Planes**GEOMETRY LESSON 1-3 (For help, go to Lesson 1-2.) Judging by appearances, will the lines intersect? 1.2.3. 4. the bottom 5. the top 6. the front 7. the back 8. the left side 9. the right side Name the plane represented by each surface of the box. 1-3**Segments, Rays, Parallel Lines and Planes**GEOMETRY LESSON 1-3 Solutions 1. no 2. yes 3. yes 4-9. Answers may vary. Samples given: 4.NMR5.PQL 6.NKL7.PQR 8.PKN9.LQR 1-3**Name the segments and rays in the figure.**The labeled points in the figure areA,B,and C. A segment is a part of a line consisting of two endpoints and all points between them. A segment is named by its two endpoints. So the segments are BA (or AB) and BC (or CB). A ray is a part of a line consisting of one endpoint and all the points of the line on one side of that endpoint. A ray is named by its endpoint first, followed by any other point on the ray. So the rays are BA and BC. Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 1-3**Use the figure below. Name all segments that**are parallel to AE. Name all segments that are skew to AE. Parallel segments lie in the same plane, and the lines that contain them do not intersect. The three segments in the figure above that are parallel to AE are BF, CG, and DH. Skew lines are lines that do not lie in the same plane. The four lines in the figure that do not lie in the same plane as AE are BC, CD, FG, and GH. Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 1-3**Identify a pair of parallel planes in your classroom.**Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 Planes are parallel if they do not intersect. If the walls of your classroom are vertical, opposite walls are parts of parallel planes. If the ceiling and floor of the classroom are level, they are parts of parallel planes. 1-3**Pages 19-23 Exercises**1. 2. 3. 4. 5.RS, RT, RW, ST, SW, TW 6.RS, ST, TW, WT, TS, SR 7.a.TS or TR, TW b.SR, ST 8. 4; RY, SY, TY, WY 9. Answers may vary. Sample: 2; YS or YR, YT or YW 10. Answers may vary. Check students’ work. 11.DF 12.BC 13.BE, CF 14. DE, EF, BE 15.AD, AB, AC 16.BC, EF 17.ABC || DEF Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 1-3**18-20Answers may vary. Samples are given**18.BE || AD 19.CF, DE 20. DEF, BC 21. FG 22. Answers may vary. Sample: CD, AB 23.BG, DH, CL 24.AF 25. true 26. False; they are skew. 27. true 28. False; they intersect above CG. 29. true 30. False; they intersect above pt. A. Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 31. False; they are ||. 32. False; they are ||. 33. Yes; both name the segment with endpoints X and Y. 34. No; the two rays have different endpoints. 35. Yes; both are the line through pts. X and Y. 1-3**36.**37. always 38. never 39. always 40. always 41. never 42. sometimes 43. always 44. sometimes 45. always 46. sometimes 47. sometimes 48. Answers may vary. Sample: (0, 0); check students’ graphs. 49.a. Answers may vary. Sample: northeast and southwest b. Answers may vary. Sample: northwest and southeast, east and west 50. Two lines can be parallel, skew, or intersecting in one point. Sample: train tracks–parallel; vapor trail of a northbound jet and an eastbound jet at different altitudes– skew; streets that cross–intersecting Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 1-3**51. Answers may vary. Sample: Skew lines cannot be contained**in one plane. Therefore, they have “escaped” a plane. 52.ST || UV 53. Answers may vary. Sample: XY and ZW intersect at R. 54. Planes ABC and DCBF intersect in BC. 55.a. The lines of intersection are parallel. b. Examples may vary. Sample: The floor and ceiling are parallel. A wall intersects both. The lines of intersection are parallel. 56. Answers may vary. Sample: The diamond structure makes it tough, strong, hard, and durable. The graphite structure makes it soft and slippery. 57.a. one segment; EF b. 3 segments; EF, EG, FG Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 1-3**57. c.**Answers may vary. Sample: For each “new” point, the number of new segments equals the number of “old” points. d. 45 segments e. 58. No; two different planes cannot intersect in more than one line. 59. yes; plane P, for example 60. Answers may vary. Sample: VR, QR, SR 61. QR 62. Yes; no; yes; explanations may vary. 63. D 64. H 65. B 66. F 67. B 68. C 69. D n(n – 1) 2 Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 1-3**70.[2]a. Alike: They do not intersect.**Different: Parallel lines are coplanar and skew lines lie in different planes. b. No; of the 8 other lines shown, 4 intersect JM and 4 are skew to JM. [1] one likeness, one difference 71–78. Answers may vary. Samples are given. 71.EF 72.A 73.C 74.AEF and HEF 75.ABH 76.EHG 77.FG 78.B Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 79. 80. 81. 82. 1.4, 1.48 83. –22, –29 84.FG, GH 85.P, S 86. No; whenever you subtract a negative number, the answer is greater than the given number. Also, if you subtract 0, the answer stays the same. 1-3**RS, TR, ST**TO, TP, TR, TS AC or BD Segments, Rays, Parallel Lines and Planes GEOMETRY LESSON 1-3 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. 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. plane BCD || plane XWQ The three pairs of lines intersect, so they cannot be parallel or skew. 1-3**Measuring Segments and Angles**GEOMETRY LESSON 1-4 (For help, go to the Skills Handbook, pages 719 and 720.) 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. 1-4**Measuring Segments and Angles**GEOMETRY LESSON 1-4 Solutions 1. The number of units from 0 to –6 on the number line is 6. 2. The number of units from 0 to 3.5 on the number line is 3.5. 3. |7 – 10| = |–3|, and the number of units from 0 to –3 on the number line is 3. 4. |–4 – 2| = |–6|, and the number of units from 0 to –6 on the number line is 6. 5. |–2 – (–4)| = |–2 + 4| = |2|, and the number of units from 0 to 2 on the number line is 2. 1-4**Measuring Segments and Angles**GEOMETRY LESSON 1-4 Solutions (continued) 6. |–3 + 12| = |9|, and the number of units from 0 to 9 on the number line is 9. 7. Combine like terms: 3x – 6 = 6; add 6 to both sides: 3x = 12; divide both sides by 3: x = 4 8. Combine like terms: 8x + 9 = 81; subtract 9 from both sides: 8x = 72; divide both sides by 8: x = 9 9. Add –7w + 2 to both sides: –6w = –2; divide both sides by –6: w = 1 3 1-4**Find which two of the segments XY, ZY, and ZW are**congruent. Because XY = ZW, XYZW. Measuring Segments and Angles GEOMETRY LESSON 1-4 Use the Ruler Postulate to find the length of each segment. XY = | –5 – (–1)| = | –4| = 4 ZY = | 2 – (–1)| = |3| = 3 ZW = | 2 – 6| = |–4| = 4 1-4**If AB = 25, find the value of x. Then find AN and NB.**AN = 2x – 6 = 2(8) – 6 = 10 NB = x + 7 = (8) + 7 = 15 Substitute 8 for x. Measuring Segments and Angles GEOMETRY LESSON 1-4 Use the Segment Addition Postulate to write an equation. AN + NB = ABSegment Addition Postulate (2x – 6) + (x + 7) = 25 Substitute. 3x + 1 = 25 Simplify the left side. 3x = 24 Subtract 1 from each side. x = 8 Divide each side by 3. AN = 10 and NB = 15, which checks because the sum of the segment lengths equals 25. 1-4**M is the midpoint of RT. Find RM, MT, and RT.**RM = MTDefinition of midpoint 5x + 9 = 8x – 36Substitute. 5x + 45 = 8xAdd 36 to each side. 45 = 3xSubtract 5x from each side. 15 = xDivide each side by 3. RM = 5x + 9 = 5(15) + 9 = 84 MT = 8x – 36 = 8(15) – 36 = 84 Substitute 15 for x. RM and MT are each 84, which is half of 168, the length of RT. Measuring Segments and Angles GEOMETRY LESSON 1-4 Use the definition of midpoint to write an equation. RT = RM + MT = 168 1-4**Name the angle below in four ways.**The name can be the number between the sides of the angle: 3. The name can be the vertex of the angle: G. Finally, the name can be a point on one side, the vertex, and a point on the other side of the angle: AGC,CGA. Measuring Segments and Angles GEOMETRY LESSON 1-4 1-4**Find the measure of each angle. Classify each as acute,**right, obtuse, or straight. Use a protractor to measure each angle. m 1 = 110 Because 90 < 110 < 180, 1 is obtuse. m 2 = 80 Because 0 < 80 < 90, 2 is acute. Measuring Segments and Angles GEOMETRY LESSON 1-4 1-4**Suppose that m 1 = 42 and m ABC = 88. Find m 2.**m 1 + m 2 = m ABCAngle Addition Postulate. 42 + m 2 = 88Substitute 42 for m 1 and 88 for m ABC. m 2 = 46 Subtract 42 from each side. Measuring Segments and Angles GEOMETRY LESSON 1-4 Use the Angle Addition Postulate to solve. 1-4