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Name ________________________________________ Date __________________ Class__________________ LESSON 2-1 Practice C Using Inductive Reasoning to Make Conjectures Make a conjecture about each pattern. Write the next two items. 1. −1, −8, −27, −64, . . . _________________________________________________________________________________________ 2. 1, 11, 21, 1211, 111221, . . . (Hint: Try reading the numbers aloud in different ways.) _________________________________________________________________________________________ 3. A, E, F, H, I, . . . _________________________________________________________________________________________ 4. _________________________________________________________________________________________ Determine if each conjecture is true. If not, write or draw a counterexample. 5. Three points that determine a plane also determine a triangle. _______________ 6. An image reflected across the x-axis cannot appear identical to its preimage. _______________ 7. If a > b and b > c, then a − b < a − c. _______________ 8. If n is an integer (n ≠ 0), then 3 1 ⎛ 1⎞ >⎜ ⎟ . n ⎝ n⎠ _______________ 9. Hank finds that a convex polygon with n sides has n − 3 diagonals from any one vertex. He notices that the diagonals from one vertex divide every polygon into triangles, and he knows that the sum of the angle measures in any triangle is 180°. Hank wants to develop a rule about the sum of the angle measures in a convex polygon with n sides. Find the rule. (Hint: Sketching some polygons may help.) _________________________________________________________________________________________ 10. A regular polygon is a polygon in which all sides have the same length and all angles have the same measure. Use your answer to Exercise 9 to determine the measure of one angle in a regular heptagon, a regular nonagon, and a regular dodecagon. Round to the nearest tenth of a degree. ____________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 2-5 Holt McDougal Geometry Answers for the chapter Geometric Reasoning 2-1 USING INDUCTIVE REASONING TO 3. The pattern is the letters of the alphabet that are made only from straight segments; K, L. MAKE CONJECTURES Practice A 1. 10 4. First rotate the figure 180°. Then reflect the figure across a vertical line. Repeat. 2. W 3. summer 5. true 6. false 4. inductive reasoning 5. true 6. even Sample answer: 7. n 7. true 8. The number of rings in a tree is the same as the tree’s age. 9. 82 rings 8. false 10. false Possible answers: n = 1, n = −1 11. Possible answers: zero, any negative number 9. Sum of angle measures = [180(n − 2)]° 10.128.6°; 140°; 150° Reteach 1 1. 1 4 12. Possible answer: Practice B 1. 36 2. 3. Arkansas 4. north 5. positive 6. n − 3 7. Possible answers: zero, any negative number 2. 36 3. 4. 5. doubled 6. 2n 7. n + 2 8. 8. Possible answer: If the lines are parallel, then they do not intersect. 9. Possible answer: If point N is between points A and G, then AN = 2 inches. 9. One-third of the bills were counterfeit. 10. Sample answer: If x = 0 and y = −1, then x 2 < y 2. 10. Each item, starting with the third, is the product of the two preceding items; 256, 8192. 11. Sample answer: ∠ABD, ∠DBE, ∠EBC, ∠ABE, ∠DBC 11. The dot skips over one vertex in a clockwise direction. 12. Sample answer: m∠1 = 25°, m∠2 = 20° 13. true Practice C 1. The pattern is the cubes of the negative integers; −125, −216. 14. Challenge 1. 1, 6, 15, 20, 15, 6, 1 2. Each item describes the item before it (one, one one, two ones, . . .); 312211, 13112221. 2. Each row has 1 as the first and last number. Each of the other numbers is found by adding the two numbers that appear just above it. Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A9 Holt McDougal Geometry