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Name _______________________________________ Date __________________ Class __________________ Lesson 2-1 (pg 74) Assgn 2-1: #1-21,25-31,35-39 Using Inductive Reasoning to Make Conjectures When you make a general rule or conclusion based on a pattern, you are using inductive reasoning. A conclusion based on a pattern is called a conjecture. Pattern 8, 3, 2, 7, . . . Conjecture Each term is 5 more than the previous term. Next Two Items 7 5 12 12 5 17 The measure of each angle is half the measure of the previous angle. Notes: Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Geometry Name _______________________________________ Date __________________ Class __________________ Find the next item in each pattern. 1. 1 1 3 , , , 1, . . . 4 2 4 2. 100, 81, 64, 49, . . . ________________________________________ 3. ________________________________________ 4. ________________________________________ ________________________________________ Complete each conjecture. 5. If the side length of a square is doubled, the perimeter of the square is __________________________________ . 6. The number of nonoverlapping angles formed by n lines intersecting in a point is __________________________________ . Use the figure to complete the conjecture in Exercise 7. 7. The perimeter of a figure that has n of these triangles is __________________________________ . Notes: Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Geometry Name _______________________________________ Date __________________ Class __________________ Lesson 2-1 cont Using Inductive Reasoning to Make Conjectures continued Since a conjecture is an educated guess, it may be true or false. It takes only one example, or counterexample, to prove that a conjecture is false. Conjecture: For any integer n, n 4n. n n 4n True or False? 3 3 4(3) 3 12 true 0 0 4(0) 00 true 2 2 4(2) 2 8 false n 2 is a counterexample, so the conjecture is false. Notes: Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Geometry Name _______________________________________ Date __________________ Class __________________ Show that each conjecture is false by finding a counterexample. 8. If three lines lie in the same plane, then they intersect in at least one point. ________________________________________________________________________________________ 9. Points A, G, and N are collinear. If AG 7 inches and GN 5 inches, then AN 12 inches. ________________________________________________________________________________________ 10. For any real numbers x and y, if x y, then x 2 y 2. ________________________________________________________________________________________ 11. The total number of angles in the figure is 3. ________________________________________ ________________________________________ 12. If two angles are acute, then the sum of their measures equals the measure of an obtuse angle. ________________________________________ ________________________________________ Determine whether each conjecture is true. If not, write or draw a counterexample. 13. Points Q and R are collinear. ________________________________________ 14. If J is between H and K, then HJ JK. ________________________________________ Notes: Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Geometry Name _______________________________________ Date __________________ Class __________________ Answers for the chapter Geometric Reasoning USING INDUCTIVE REASONING TO MAKE CONJECTURES Practice A 1. 10 2. W 3. summer 3. The pattern is the letters of the alphabet that are made only from straight segments; K, L. 4. First rotate the figure 180. Then reflect the figure across a vertical line. Repeat. 5. true 6. false 4. inductive reasoning 5. true 6. even Sample answer: 7. n 8. The number of rings in a tree is the same as the tree’s age. 9. 82 rings 7. true 8. false Possible answers: n 1, n 1 10. false 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 2. 36 3. 4. 7. Possible answers: zero, any negative number 5. doubled 6. 2n 8. 8. Possible answer: If the lines are parallel, then they do not intersect. 9. One-third of the bills were counterfeit. 10. Each item, starting with the third, is the product of the two preceding items; 256, 8192. 11. The dot skips over one vertex in a clockwise direction. Practice C 1. The pattern is the cubes of the negative integers; 125, 216. 2. Each item describes the item before it (one, one one, two ones, . . .); 312211, 13112221. 7. n 2 9. Possible answer: If point N is between points A and G, then AN 2 inches. 10. Sample answer: If x 0 and y 1, then x 2 y 2. 11. Sample answer: ABD, DBE, EBC, ABE, DBC 12. Sample answer: m1 25°, m2 20° 13. true 14. Challenge 1. 1, 6, 15, 20, 15, 6, 1 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. Holt McDougal Geometry