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NAME DATE 2-1 PERIOD Study Guide and Intervention Inductive Reasoning and Conjecture Example 1 Write a conjecture about the next number in the sequence 1, 3, 9, 27, 81. Example 2 Write a conjecture about the number of small squares in the next figure. Look for a pattern: The sides of the squares have measures 1, 2, and 3 units. Conjecture: For the next figure, the side of the square will be 4 units, so the figure will have 16 small squares. Look for a pattern: Each number is a power of 3. 1 3 9 27 81 0 1 3 3 32 33 34 Conjecture: The next number will be 35 or 243. Lesson 2-1 Making Conjectures Inductive reasoning is reasoning that uses information from different examples to form a conclusion or statement called a conjecture. Exercises Write a conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. 1. -5, 10, -20, 40 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2. 1, 10, 100, 1000 6 7 8 3. 1, − , −, − 5 5 5 Write a conjecture about each value or geometric relationship. 4. A(-1, -1), B(2, 2), C(4, 4) 5. ∠1 and ∠2 form a right angle. 6. ∠ABC and ∠DBE are vertical angles. 7. ∠E and ∠F are right angles. Chapter 2 005_054_GEOCRMC02_890511.indd 5 5 Glencoe Geometry PDF Pass 4/9/08 5:13:32 PM NAME 2-1 DATE Study Guide and Intervention PERIOD (continued) Inductive Reasoning and Conjecture Find Counterexamples A conjecture is false if there is even one situation in which the conjecture is not true. The false example is called a counterexample. Example Find a counterexample to show the conjecture is false. −− −− −− If AB " BC, then B is the midpoint of AC. −− −− Is it possible to draw a diagram with AB " BC such that B is not the midpoint? This diagram is a counterexample because −− point B is not on AC. The conjecture is false. C A 3 cm B 3 cm Exercises Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 1. If points A, B, and C are collinear, then AB + BC = AC. Chapter 2 005_054_GEOCRMC02_890511.indd 6 −−− −− 4. If DE ⊥ EF, then ∠ DEF is a right angle. 6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3. If ∠ ABC and ∠ DEF are supplementary, then ∠ ABC and ∠ DEF form a linear pair. 2. If ∠R and ∠ S are supplementary, and ∠ R and ∠ T are supplementary, then ∠ T and ∠ S are congruent. Glencoe Geometry PDF Pass 4/9/08 5:13:36 PM NAME 2-1 DATE PERIOD Skills Practice Inductive Reasoning and Conjecture Write a conjecture that describes the pattern in the sequence. Then use your conjecture to find the next item in the sequence. 1. 9 11 , 5, − ,4 3. 6, − 2 2 4. -2, 4, -8, 16, -32 Lesson 2-1 2. -4, -1, 2, 5, 8 Make a conjecture about each value or geometric relationship. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. Points A, B, and C are collinear, and D is between B and C. 7. ∠1, ∠2, ∠3, and ∠4 form four linear pairs. −−− 6. Point P is the midpoint of NQ 8. ∠3 # ∠4 Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 9. If ∠ ABC and ∠ CBD form a linear pair, then ∠ ABC # ∠ CBD. −− −− −− 10. If AB, BC, and AC are congruent, then A, B, and C are collinear. 11. If AB + BC = AC, then AB = BC. 12. If ∠1 is complementary to ∠2, and ∠1 is complementary to ∠3, then ∠2 # ∠3. Chapter 2 005_054_GEOCRMC02_890511.indd 7 7 Glencoe Geometry PDF Pass 5/13/08 5:53:46 PM NAME 2-1 DATE PERIOD Practice Inductive Reasoning and Conjecture Make a conjecture about the next item in each sequence. 1. 2. 5, -10, 15, -20 1 1 1 3. -2, 1, - − , −, - − 2 4 8 4. 12, 6, 3, 1.5, 0.75 Make a conjecture about each value or geometric relationship. 5. ∠ABC is a right angle. 6. Point S is between R and T. 7. P, Q, R, and S are noncollinear −−− −−− −− −− and PQ # QR # RS # SP. 8. ABCD is a parallelogram. 10. If ∠1 and ∠2 are adjacent angles, then ∠1 and ∠2 form a linear pair. −−− −−− −− −− 11. If GH and JK form a right angle and intersect at P, then GH ⊥ JK. 12. ALLERGIES Each spring, Rachel starts sneezing when the pear trees on her street blossom. She reasons that she is allergic to pear trees. Find a counterexample to Rachel’s conjecture. Chapter 2 005_054_GEOCRMC02_890511.indd 8 8 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. −−− 9. If S, T, and U are collinear and ST = TU, then T is the midpoint of SU. Glencoe Geometry PDF Pass 4/9/08 5:13:44 PM