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Name Date Class Reteach LESSON 2-1 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 Conjecture 8, 3, 2, 7, . . . Next Two Items 7 5 12 12 5 17 Each term is 5 more than the previous term. 45° The measure of each angle is half the measure of the previous angle. Find the next item in each pattern. 3, 1, . . . 1 , __ 1, __ 1. __ 4 2 4 11.25° 22.5° 2. 100, 81, 64, 49, . . . 1 1__ 4 36 3. 4. 3 6 10 Complete each conjecture. 5. If the side length of a square is doubled, the perimeter of the square is doubled . 6. The number of nonoverlapping angles formed by n lines intersecting in a point is 2n . Use the figure to complete the conjecture in Exercise 7. 7. The perimeter of a figure that has n of these triangles is 1 1 n2 . 1 1 03 Copyright © by Holt, Rinehart and Winston. All rights reserved. 6 1 1 1 1 04 1 1 1 1 1 1 05 1 1 1 1 06 Holt Geometry Name LESSON 2-1 Date Class Reteach 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. 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. Possible answer: If the lines are parallel, then they do not intersect. 9. Points A, G, and N are collinear. If AG 7 inches and GN 5 inches, then AN 12 inches. Possible answer: If point N is between points A and G, then AN 2 inches. 10. For any real numbers x and y, if x y, then x 2 y 2. 2 2 Sample answer: If x 0 and y 1, then x y . 11. The total number of angles in the figure is 3. $ Sample answer: ABD, DBE, " ! EBC, ABE, DBC % # 12. If two angles are acute, then the sum of their measures equals the measure of an obtuse angle. Sample answer: m1 25, m2 20 Determine whether each conjecture is true. If not, write or draw a counterexample. 14. If J is between H and K, then HJ JK. 13. Points Q and R are collinear. ( true Copyright © by Holt, Rinehart and Winston. All rights reserved. 7 * + Holt Geometry Name LESSON 2-1 Date Class Name Practice A 2-1 Find the next item in each pattern. Using Inductive Reasoning to Make Conjectures Find the next item in each pattern. 2. Z, Y, X, . . . 1. 2, 4, 6, 8, . . . Class Practice B LESSON Using Inductive Reasoning to Make Conjectures 3. fall, winter, spring, . . . � 5. A statement you believe to be 3. Alabama, Alaska, Arizona, . . . . true north Complete each conjecture. 5. The square of any negative number is For Exercises 6–8, complete each conjecture by looking for a pattern in the examples. even positive . 6. The number of diagonals that can be drawn from one vertex in a convex polygon . n�3 that has n vertices is n 7. The number of sides of a polygon that has n vertices is ������� 4. west, south, east, . . . Arkansas based on inductive reasoning is called a conjecture. 6. The sum of two odd numbers is 3�5�8 13 � 3 � 16 1�1�2 � 36 4. When several examples form a pattern and you assume the pattern will continue, inductive reasoning 2. 1. 100, 81, 64, 49, . . . summer W 10 you are applying Date . Show that each conjecture is false by finding a counterexample. . 7. For any integer n, n 3 � 0. Possible answers: zero, any negative number 8. Each angle in a right triangle has a different measure. 8. When a tree is cut horizontally, a series of rings is visible in the stump. Make a conjecture about the number of rings and the age of the tree based on the data in the table. Number of Rings 3 15 22 60 Age of Tree (years) 3 15 22 60 ��� 9. For many years in the United States, each bank printed its own currency. The variety of different bills led to widespread counterfeiting. By the time of the Civil War, a significant fraction of the currency in circulation was counterfeit. If one Civil War soldier had 48 bills, 16 of which were counterfeit, and another soldier had 39 bills, 13 of which were counterfeit, make a conjecture about what fraction of bills were counterfeit at the time of the Civil War. The number of rings in a tree is the same as the tree’s age. 9. Assume your conjecture in Exercise 8 is true. Find the number of rings in an 82-year-old oak tree. 82 rings false 10. A counterexample shows that a conjecture is One-third of the bills were counterfeit. . Make a conjecture about each pattern. Write the next two items. Show that each conjecture is false by finding a counterexample. 10. 1, 2, 2, 4, 8, 32, . . . 11. For any number n, 2n � n. Possible answers: zero, any negative number 12. Two rays having the same endpoint make an acute angle. (Sketch a counterexample.) 3 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name LESSON 2-1 11. � Each item, starting with the Possible answer: Date Holt Geometry Class The dot skips over one vertex in a clockwise direction. 4 2-1 Using Inductive Reasoning to Make Conjectures The pattern is the cubes of the negative integers; �125, �216. Pattern Conjecture Each item describes the item before it (one, one one, two ones, . . .); 312211, 13112221. 3. A, E, F, H, I, . . . 45° The pattern is the letters of the alphabet that are made only from straight segments; K, L. � 36 4. 3 true 6 10 �� false � Sample answer: Complete each conjecture. n 3 �n � 8. If n is an integer (n � 0), then _1_ � _1_ 5. If the side length of a square is doubled, the perimeter of the square true 7. If a � b and b � c, then a � b � a � c. is false Possible answers: n � 1, n � �1 . 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.) Copyright © by Holt, Rinehart and Winston. All rights reserved. . is 2n . Use the figure to complete the conjecture in Exercise 7. 7. The perimeter of a figure that has n of these triangles 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. 128.6�; 140�; 5 doubled 6. The number of nonoverlapping angles formed by n lines intersecting in a point Sum of angle measures � [180(n � 2)]� Copyright © by Holt, Rinehart and Winston. All rights reserved. 2. 100, 81, 64, 49, . . . 3. Determine if each conjecture is true. If not, write or draw a counterexample. 6. An image reflected across the x-axis cannot appear identical to its preimage. � 11.25° 22.5° 1_1_ 4 First rotate the figure 180�. Then reflect the figure across a vertical line. Repeat. 5. Three points that determine a plane also determine a triangle. 7 � 5 � 12 12 � 5 � 17 The measure of each angle is half the measure of the previous angle. Find the next item in each pattern. 1. _1_, _1_, _3_, 1, . . . 4 2 4 , �������� Next Two Items Each term is 5 more than the previous term. �8, �3, 2, 7, . . . 2. 1, 11, 21, 1211, 111221, . . . (Hint: Try reading the numbers aloud in different ways.) � Holt Geometry Class 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. 1. �1, �8, �27, �64, . . . � Date Reteach LESSON Using Inductive Reasoning to Make Conjectures ������� preceding items; 256, 8192. Name Practice C � , third, is the product of the two Copyright © by Holt, Rinehart and Winston. All rights reserved. Make a conjecture about each pattern. Write the next two items. 4. ��� is n�2 . 1 1 1 ��3 1 1 1 1 ��4 1 1 1 1 1 ��5 1 1 1 1 1 1 ��6 150� Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 59 6 Holt Geometry Holt Geometry Name LESSON 2-1 Date Class Name Reteach Date Challenge LESSON Using Inductive Reasoning to Make Conjectures 2-1 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. Class Discovery Through Patterns The pattern shown is known as Pascal’s Triangle. 1. If the pattern is extended, find the terms in row 7. Conjecture: For any integer n, n � 4n. � ����� 1, 6, 15, 20, 15, 6, 1 � ����� n n � 4n True or False? 3 3 � 4(3) 3 � 12 true 0 0 � 4(0) 0�0 true number. Each of the other numbers is ����� �2 �2 � 4(�2) �2 � �8 false found by adding the two numbers that ����� 2. Make a conjecture for the pattern. � ����� Each row has 1 as the first and last � ����� � � � � � � � � � � �� � � �� � � � appear just above it. n � �2 is a counterexample, so the conjecture is false. 3. Make a conjecture about the sum of the terms in each row. Show that each conjecture is false by finding a counterexample. The sum of each row of terms after the first is twice the sum of the terms in the previous row. 8. If three lines lie in the same plane, then they intersect in at least one point. Possible answer: If the lines are parallel, then they do not intersect. Refer to the pattern of figures for Exercises 4 and 5. 9. Points A, G, and N are collinear. If AG � 7 inches and GN � 5 inches, then AN � 12 inches. Possible answer: If point N is between points A and G, then AN � 2 inches. 2 Figure 1 2 10. For any real numbers x and y, if x � y, then x � y . Sample answer: If x � 0 and y � �1, then x 2 � y 2. 11. The total number of angles in the figure is 3. � � 27; 81 5. Write an algebraic expression for the number of black triangles in figure n. 3 2 6. For every integer x, x � 2x � 1 is divisible by 2. 12. If two angles are acute, then the sum of their measures equals the measure of an obtuse angle. Sample answer: x � 2 Sample answer: n � �2 2 7. For every integer n, n � n is prime. a 8. Make a table of values for the expression 4 � 1, where a is a positive integer. Make a conjecture about the type of number that is generated by the rule. Sample answer: m�1 � 25�, m�2 � 20� 13. Points Q and R are collinear. 14. If J is between H and K, then HJ � JK. � true 7 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name � Date � Class a a 1 2 3 4 5 Determine whether each conjecture is true. If not, write or draw a counterexample. 2-1 Holt Geometry Problem Solving 4 �1 3 15 63 255 1023 1. Estimate the length of a green iguana after 1 year if it was 8 inches long when it hatched. Length after Hatching (in.) Length after 1 Year (in.) 1 10 36 2 9 34 3 11 35 4 12 35 5 10 37 Sample answer: 33 in. 2. Make a conjecture about the average growth of a green iguana during the first year. 2 3 4 5 6 7 ������������������� ������������������ ���������� ����������� ��������� ����������� 8 5. The table shows the number of cells present during three phases of mitosis. If a sample contained 80 cells during interphase, which is the best prediction for the number of cells present during prophase? A 18 cells B 24 cells Match 4 was 18 minutes long. Sample C 40 cells D 80 cells 0 Number of Cells Interphase Prophase 86 22 5 2 70 28 3 3 76 32 3 4 91 25 5 5 65 16 4 6 89 34 6 H 306 students A $20.50 C $24.50 G 204 students J 333 students B $23.33 D $25.00 9 Copyright © by Holt, Rinehart and Winston. All rights reserved. 1 2 3 4 5 6 � 2. Consider the two conjectures given to explain the pattern seen on the grid. Metaphase 1 F 102 students (4, 4) (3, 3) (2, 2) (1, 1) 1. Observe the pattern of the points on the grid. Conjecture A or To find the next point on the grid, add 1 to the x-coordinate and subtract 1 from the y-coordinate. Conjecture B To find the next point on the grid, add 1 to both the x-coordinate and the y-coordinate. 3. Evaluate each of the conjectures and plot the next two points on the graph according to each conjecture. Which conjecture is correct? 7. Mara earned $25, $25, $20, and $28 in the last 4 weeks for walking her neighbor’s dogs. If her earnings continue in this way, which is the best estimate for her average weekly earnings for next month? 6. About 75% of the students at Jackson High School volunteer to clean up a half-mile stretch of road every year. If there are 408 students in the school this year, about how many are expected to volunteer for the clean-up? Copyright © by Holt, Rinehart and Winston. All rights reserved. � 6 5 4 3 2 1 4. These matches were all longer than a half hour. Match 7 lasted 1 hour 3 minutes. Choose the best answer. ������ ��������������� Consider the following example: 0:31 0:56 0:51 0:18 0:50 0:34 1:03 0:36 3. Every one of the first eight matches lasted less than 1 hour. Holt Geometry Identify Relationships The times for the first eight matches of the Santa Barbara Open women’s volleyball tournament are shown. Show that each conjecture is false by finding a counterexample. 1 Class The steps to making a decision based on inductive reasoning are presented below. Sample answer: The average growth of a green iguana during the first year is about 2 ft. Time Date Reading Strategies 2-1 Iguana 8 Copyright © by Holt, Rinehart and Winston. All rights reserved. LESSON The table shows the lengths of five green iguanas after birth and then after 1 year. Sample answer: All values of a 4 � 1 are divisible by 3. Name Using Inductive Reasoning to Make Conjectures Match n�1 Find a counterexample for each statement. � �EBC, �ABE, �DBC LESSON Figure 3 � Sample answer: �ABD, �DBE, � Figure 2 4. If the pattern continues, how many black triangles will there be in Figure 4? in Figure 5? For Conjecture A, students should plot (5, 3) and (6, 2). For Conjecture B, students should plot (5, 5) and (6, 6). Student should conclude that Conjecture B is correct. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 60 10 Holt Geometry Holt Geometry