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

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.

# of sides of polygon Name of Polygon# of diagonals from 1 vertex # of triangles formed sum of angle measures n triangle º quadrilateral º pentagon º hexagon º septagon º octagon º nonagon º decagon º undecagon º dodecagon º n-gon n - 3n - 2 (n-2)  180º

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

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

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

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

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

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

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

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.

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

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 …

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.

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

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.

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

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

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

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 $ = $25 3 x $9 = $27 $ (bellhop) = $25

Patterns in Mathematics 1 x = 9 12 x = x = x = x = x = x = x = x =

1 x = x = x = x = x = x = x = x = x 9 +10= Page 108 #15

9 x = x = x = x = x = x = x = x = Patterns in Mathematics Page 108 #16

9 2 = = = = = = = = = Patterns in Mathematics Page 108 #17

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?