1. Lesson 3-6 Inductive Reasoning (page 106) Essential Question How can you apply parallel lines (planes) to make deductions?

2. INDUCTIVE REASONING: a kind of reasoning in which the conclusion is based on several past observations. Note: The conclusion is probably true, but not necessarily true.

3. # of sides of polygon Name of Polygon# of diagonals from 1 vertex # of triangles formed sum of angle measures 3 4 5 6 7 8 9 10 11 12 n triangle 0 1 180º quadrilateral 1 2 360º pentagon 2 3 540º hexagon 3 4 720º septagon 4 5 900º octagon 5 6 1080º nonagon 6 7 1260º decagon 7 8 1440º undecagon 8 9 1620º dodecagon 9 10 1800º n-gon n - 3n - 2 (n-2)  180º

4. Example # 1. On each of the first 6 days Noah attended his geometry class, Mrs. Heller, his geometry teacher, gave a homework assignment. Noah concludes that he will have geometry homework every _______ he has geometry class. day

5. Example # 2. (a) Look for a pattern and predict the next two numbers or letters. 1, 3, 7, 13, 21, ____, ____ 31 43

6. Example # 2. (b) Look for a pattern and predict the next two numbers or letters. 81, 27, 9, 3, ____, ____ 1 1/3

7. Example # 2. (c) Look for a pattern and predict the next two numbers or letters. 3, - 6, 12, - 24, ____, ____ 48 - 96

8. Example # 2. (d) Look for a pattern and predict the next two numbers or letters. 1, 1, 2, 3, 5, 8, 13, 21, ____, ____ 34 55

9. Example # 2. (e) Look for a pattern and predict the next two numbers or letters. O, T, T, F, F, S, S, ____, ____ E N

10. Example # 2. (f) Look for a pattern and predict the next two numbers or letters. J, M, M, J, ____, ____ S N

11. DEDUCTIVE REASONING: proving statements by reasoning from accepted postulates, definitions, theorems, and given information. Note: The conclusion must be true if the hypotheses are true.

12. Example # 3. In the same geometry class, Hannah reads the theorem, “Vertical angles are congruent.” She notices in a diagram that angle 1 and angle 2 are vertical angles. Hannah concludes that ______________. ∠ 1  ∠∠ 22

13. Example # 4. (a) Accept the two statements as given information. State a conclusion based on deductive reasoning. If no conclusion can be reached, write no conclusion. All cows eat grass. Blossom eats grass. No conclusion … Blossom could be a rabbit, goat, or …

14. Example # 4. (b) Accept the two statements as given information. State a conclusion based on deductive reasoning. If no conclusion can be reached, write no conclusion. Aaron is taller than Alex. Alex is taller than Emily. Aaron is taller than Emily.

15. Example # 4. (c) Accept the two statements as given information. State a conclusion based on deductive reasoning. If no conclusion can be reached, write no conclusion. ∠ A  ∠ B and m ∠ A = 72º m ∠ B = 72º

16. Example # 4. (d) Accept the two statements as given information. State a conclusion based on deductive reasoning. If no conclusion can be reached, write no conclusion. No conclusion … except in 2-D, but in 3-D, the lines could be skew.

17. Example # 5. (a) Tell whether the reasoning process is deductive or inductive. Aaron did his assignment and found the sums of the exterior angles of several different polygons. Noticing the results were all the same, he concludes that the sum of the measures of the exterior angles of any polygon is 360º. deductive or inductive

18. Example # 5. (b) Tell whether the reasoning process is deductive or inductive. Tammy is told that m ∠ A = 150º and m ∠ B = 30º. Since she knows the definition of supplementary angles, she concludes that ∠ A and ∠ B are supplementary. deductive or inductive

19. Example # 5. (c) Tell whether the reasoning process is deductive or inductive. Nicholas observes that the sum of 2 and 4 is an even number, that the sum of 4 and 6 is an even number, and that the sum of 12 and 6 is also an even number. He concludes that the sum of two even numbers is always an even number. deductive or inductive

20. Problem: Three businessmen stay at a hotel. The hotel room costs $30, therefore, each pays $10. The owner recalls that they get a discount. The total should be $25. The owner tells the bellhop to return $5. The bellhop decides to keep $2 and return $1 to each businessman. Now, each businessman paid $9, totaling $27, plus the $2 the bellhop kept, totaling $29. Where is the other dollar? There is no extra dollar! They paid $30 - 5 = $25 3 x $9 = $27 $27 - 2 (bellhop) = $25

26. Patterns in Mathematics 1 x 8 + 1 = 9 12 x 8 + 2 = 98 123 x 8 + 3 = 987 1234 x 8 + 4 = 9876 12345 x 8 + 5 = 98765 123456 x 8 + 6 = 987654 1234567 x 8 + 7 = 9876543 12345678 x 8 + 8 = 98765432 123456789 x 8 + 9 = 987654321

27. 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 +10= 1111111111 Page 108 #15

28. 9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888 Patterns in Mathematics Page 108 #16

29. 9 2 = 81 99 2 = 9801 999 2 = 998001 9999 2 = 99980001 99999 2 = 9999800001 999999 2 = 999998000001 9999999 2 = 99999980000001 99999999 2 = 9999999800000001 999999999 2 = 999999998000000001 Patterns in Mathematics Page 108 #17

30. Assignment Written Exercises on pages 107 & 108 RECOMMENDED: 15 to 25 odd #’s REQUIRED: 1 to 13 ALL #’s Prepare for a test on Chapter 3: Parallel Lines and Planes How can you apply parallel lines (planes) to make deductions?