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Lesson 2.1 • Inductive Reasoning Name Period Date For Exercises 1–8, use inductive reasoning to find the next two terms in each sequence. 1. 4, 8, 12, 16, _____, _____ 2. 400, 200, 100, 50, 25, _____, _____ 1 2 1 4 8 7 2 5 4. 5, 3, 2, 1, 1, 0, _____, _____ 3. , , , , _____, _____ 5. 360, 180, 120, 90, _____, _____ 6. 1, 3, 9, 27, 81, _____, _____ 7. 1, 5, 17, 53, 161, _____, _____ 8. 1, 5, 14, 30, 55, _____, _____ For Exercises 9–12, use inductive reasoning to draw the next two shapes in each picture pattern. 9. 10. 11. 12. y y y (–1, 3) (3, 1) x x x (–3, –1) For Exercises 13–15, use inductive reasoning to test each conjecture. Decide if the conjecture seems true or false. If it seems false, give a counterexample. 13. Every odd whole number can be written as the difference of two squares. 14. Every whole number greater than 1 can be written as the sum of two prime numbers. 15. The square of a number is larger than the number. Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press CHAPTER 2 9 Lesson 2.2 • Deductive Reasoning Name Period Date 1. ABC is equilateral. Is ABD equilateral? What type of reasoning, B inductive or deductive, do you use when solving this problem? 1 2. If 6 # 8 7, 10 # 3 6, and 3 # 2 2.5, then 2 4 # 8 _____ 5 # 0 _____ 2 # 2 _____ C D A What type of reasoning, inductive or deductive, do you use when solving this problem? 3. A and D are complementary. A and E are supplementary. What can you conclude about D and E? What type of reasoning, inductive or deductive, do you use when solving this problem? 4. g. a. d. e. b. f. c. Whatnots Not whatnots Which are whatnots? What type of reasoning, inductive or deductive, do you use when solving this problem? 5. Solve each equation for x. Give a reason for each step in the process. 19 2(3x 1) 5 What type of reasoning, inductive or deductive, do you use when solving these problems? a. 4x 3(2 x) 8 2x b. x 2 6. A sequence is generated by the function f(n) 5 n 2. Give the first five terms in the sequence. What type of reasoning, inductive or deductive, do you use when solving this problem? 7. A sequence begins 4, 1, 6, 11 . . . a. Give the next two terms in the sequence. What type of reasoning, inductive or deductive, do you use when solving this problem? b. Find a rule that generates the sequence. Then give the 50th term in the sequence. What type of reasoning, inductive or deductive, do you use when solving this problem? 8. Choose any 3-digit number. Multiply it by 7. Multiply the result by 11. Then multiply the result by 13. Do you notice anything? Try a few other 3-digit numbers and make a conjecture. Use deductive reasoning to explain why your conjecture is true. 10 CHAPTER 2 Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press Lesson 2.3 • Finding the nth Term Name Period Date For Exercises 1–4, tell whether or not the rule is a linear function. 1. n 1 2 3 4 5 f(n) 8 15 22 29 36 3. n h(n) 2. n g(n) 1 2 3 4 5 9 6 2 3 9 4. 1 2 3 4 5 14 11 8 5 2 1 2 n j(n) 3 4 5 32 1 12 0 1 2 For Exercises 5 and 6, complete each table. 5. n 1 2 3 4 5 6. n 6 f(n) 7n 12 1 2 3 4 5 6 g(n) 8n 2 For Exercises 7–9, find the function rule for each sequence. Then find the 50th term in the sequence. 7. n f(n) 8. n g(n) 9. n h(n) 1 2 3 4 5 6 9 13 17 21 25 29 1 2 3 4 5 6 6 1 4 9 1 2 3 4 5 6 6.5 7 7.5 8 8.5 9 ... n ... 50 ... n ... 50 ... n ... 50 14 19 10. Find the rule for the number of tiles in the nth figure. Then find the number of tiles in the 200th figure. n 1 2 3 Number of tiles 1 4 7 4 5 ... n ... 200 11. Sketch the next figure in the sequence. Then complete the table. Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press n 1 2 Number of segments and lines 2 6 Number of regions of the plane 4 3 4 ... n ... CHAPTER 2 50 11 Lesson 2.4 • Mathematical Modeling Name Period Date 1. If you toss a coin, you will get a head or a tail. Copy and complete the geometric model to show all possible results of four consecutive tosses. H H T How many sequences of results are possible? How many sequences have exactly one tail? Assuming a head or a tail is equally likely, what is the probability of getting exactly one tail in four tosses? 2. If there are 12 people sitting around a table, how many different pairs of people can have conversations during dinner, assuming they can all talk to each other? What geometric figure can you use to model this situation? 3. Tournament games and results are often displayed using a geometric model. Two examples are shown below. Sketch a geometric model for a tournament involving 4 teams and a tournament involving 5 teams. Each team must have the same chance to win. Try to have as few games as possible in each tournament. Show the total number of games in each tournament. Name the teams a, b, c . . . and number the games 1, 2, 3 . . . . a a 1 b 3 1 3 c b 2 3 teams, 3 games (round robin) 12 CHAPTER 2 c d 2 4 teams, 3 games (single elimination) Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press Lesson 2.5 • Angle Relationships Name Period Date For Exercises 1–8, find each lettered angle measure without using a protractor. 1. 2. a 112° 3. a 15° b c 4. e 132° d 6. b 7. c c a b c 8. a b ba 42° d a 70° 5. 66° 40° 38° c 70° d e 110° a b 138° 100° a b 25° For Exercises 9–14, tell whether each statement is always (A), sometimes (S), or never (N) true. 9. _____ The sum of the measures of two acute angles equals the measure of an obtuse angle. 10. _____ If XAY and PAQ are vertical angles, then either X, A, and P or X, A, and Q are collinear. 11. _____ The sum of the measures of two obtuse angles equals the measure of an obtuse angle. 12. _____ The difference between the measures of the supplement and the complement of an angle is 90°. 13. _____ If two angles form a linear pair, then they are complementary. 14. _____ If a statement is true, then its converse is true. For Exercises 15–19, fill in each blank to make a true statement. 15. If one angle of a linear pair is obtuse, then the other is ____________. 16. If A B and the supplement of B has measure 22°, then mA ________________. 17. If P is a right angle and P and Q form a linear pair, then mQ is ________________. 18. If S and T are complementary and T and U are supplementary, then U is a(n) ________________ angle. 19. Switching the “if ” and “then” parts of a statement changes the statement to its ________________. Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press CHAPTER 2 13 Lesson 2.6 • Special Angles on Parallel Lines Name Period Date For Exercises 1–11, use the figure at right. For Exercises 1–5, find an example of each term. 1. Corresponding angles 2. Alternate interior angles 3. Alternate exterior angles 4. Vertical angles 3 4 7 8 1 2 5 6 5. Linear pair of angles For Exercises 6–11, tell whether each statement is always (A), sometimes (S), or never (N) true. 6. _____ 1 3 7. _____ 3 8 8. _____ 2 and 6 are supplementary. 9. _____ 7 and 8 are supplementary. 10. _____ m1 m6 11. _____ m5 m4 For Exercises 12–14, use your conjectures to find each angle measure. 12. 13. 14. 54° a a a 65° b b 54° c d c b For Exercises 15–17, use your conjectures to determine whether or not 1 2, and explain why. If not enough information is given, write “cannot be determined.” 15. 16. 118° 62° 17. 1 1 2 18. Find each angle measure. 48° 95° 48° 2 25° 1 2 44° f 78° e 64° d c a b 14 CHAPTER 2 Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press LESSON 1.7 • A Picture Is Worth a Thousand Words 1. 8. Possible answer: 9. Possible answer: Possible locations Gas 5 10 m A 12 m 5 Power 2. 30 m 11. x 2, y 1 10. 18 cubes Wall 4m 12 m Station 1 LESSON 2.1 • Inductive Reasoning Station 2 12 m Possible locations 4m Wall 3. Dora, Ellen, Charles, Anica, Fred, Bruce 1 1 3 1. 20, 24 1 1 2. 122, 64 5 3. 4, 2 4. 1, 1 5. 72, 60 6. 243, 729 7. 485, 1457 8. 91, 140 9. 4 3 10. D E C A F 11. B 4. Possible answers: a. b. c. 12. y y (3, 1) x x (1, –3) 13. True LESSON 1.8 • Space Geometry 1. 2. 14. False; 11 is not a sum of any two prime numbers. 1 2 1 15. False; 2 4 LESSON 2.2 • Deductive Reasoning 3. 2. 6, 2.5, 2; inductive 1. No; deductive 3. mE mD (mE mD 90°); deductive 4. a, e, f; inductive 5. Deductive 4. Rectangular prism 5. Pentagonal prism a. 4x 3(2 x) 8 2x 4x 6 3x 8 2x x 6 8 2x 3x 6 8 6. 7. 3x 2 2 x 3 Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press The original equation. Distributive property. Combining like terms. Addition property of equality. Subtraction property of equality. Division property of equality. ANSWERS 89 19 2(3x 1) b. 5 x 2 The original equation. 19 2(3x 1) 5(x 2) Multiplication property 11. (See table at bottom of page.) of equality. 19 6x 2 5x 10 Distributive property. 21 6x 5x 10 Combining like terms. 21 11x 10 Addition property of equality. 11 11x LESSON 2.4 • Mathematical Modeling Subtraction property of equality. 1x 1. H Division property of equality. H T 6. 4, 1, 4, 11, 20; deductive H H 7. a. 16, 21; inductive b. f(n) 5n 9; 241; deductive T T 8. Sample answer: If any 3-digit number “XYZ” is multiplied by 7 11 13, then the result will be of the form “XYZ,XYZ.” This is because 7 11 13 1001. For example, 451 H H T T 7 11 13 451(7 11 13) H T 451(1001) T 451(1000 1) LESSON 2.3 • Finding the nth Term 2. Linear T H T H H T H T H T H T H T 3. Not linear 3. Possible answers: b 4. Linear 5. n 1 1 f (n) 2 5 3 2 4 9 5 16 6 23 30 1 2 3 4 5 6 g(n) 10 18 26 34 42 50 3 4 4 5 6 f 6 teams, 7 games 5 teams, 10 games a b 1 8. f(n) 5n 11; f(50) 239 1 9. f(n) 2n 6; f(50) 31 10. 2 7 d 4 e 7. f(n) 4n 5; f(50) 205 1 c 3 e d 8 10 5 2 a 7 3 9 1 6 n n c 2 b a 6. Sequences with exactly one tail T 16 sequences of results. 4 sequences have exactly 4 1 one tail. So, P(one tail) 1 64 2. 66 different pairs. Use a dodecagon showing sides and diagonals. 451,000 451 451,451 1. Linear H f 6 4 3 5 e 2 5 n 200 c d Number of tiles 1 4 7 10 13 3n 2 6 teams, 6 games 598 Lesson 2.3, Exercise 11 11. 90 Figure number 1 2 3 4 n 50 Number of segments and lines 2 6 10 14 4n 2 198 Number of regions of the plane 4 12 20 28 8n 4 396 ANSWERS Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press LESSON 2.5 • Angle Relationships LESSON 3.1 • Duplicating Segments and Angles 1. a 68°, b 112°, c 68° 2. a 127° 1. P Q 3. a 35°, b 40°, c 35°, d 70° R S 4. a 24°, b 48° 5. a 90°, b 90°, c 42°, d 48°, e 132° A 6. a 20°, b 70°, c 20°, d 70°, e 110° 7. a 70°, b 55°, c 25° 8. a 90°, b 90° 9. Sometimes 10. Always 11. Never 13. Never 14. Sometimes 15. acute 16. 158° 17. 90° B 2. XY 3PQ 2RS X Y 12. Always 18. obtuse 3. Possible answer: 128° 35° 93° 4. B 19. converse LESSON 2.6 • Special Angles on Parallel Lines C 1. One of: 1 and 3; 5 and 7; 2 and 4; 6 and 8 5. 6. B D D 2. One of: 2 and 7; 3 and 6 3. One of: 1 and 8; 4 and 5 C 4. One of: 1 and 6; 3 and 8; 2 and 5; 4 and 7 5. One of: 1 and 2; 3 and 4; 5 and 6; 7 and 8; 1 and 5; 2 and 6; 3 and 7; 4 and 8 6. Sometimes 9. Always 7. Always 10. Never D D C 7. Four possible triangles. One is shown below. 8. Always 11. Sometimes 12. a 54°, b 54°, c 54° 13. a 115°, b 65°, c 115°, d 65° 14. a 72°, b 126° 16. 1 2 15. 1 2 17. cannot be determined 18. a 102°, b 78°, c 58°, d 122°, e 26°, f 58° Discovering Geometry Practice Your Skills ©2003 Key Curriculum Press LESSON 3.2 • Constructing Perpendicular Bisectors 1. 2. Square ANSWERS 91

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