Transcript Chapter 1 Points, Lines, Planes, and Angles page 1

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
but not necessarily
true .
true ,
# of sides of
polygon
3
4
5
6
7
8
9
10
11
12
n
Name of Polygon
triangle
# of diagonals
from 1 vertex
# of triangles
formed
sum of angle
measures
0
1
180º
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-3
n-2
quadrilateral
n-gon
(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
day he has geometry class.
every _______
Example # 2. (a)
Look for a pattern and predict the next two
numbers or letters.
31 ____
43
1, 3, 7, 13, 21, ____,
Example # 2. (b)
Look for a pattern and predict the next two
numbers or letters.
1 ____
1/3
81, 27, 9, 3, ____,
Example # 2. (c)
Look for a pattern and predict the next two
numbers or letters.
48 -____
96
3, - 6, 12, - 24, ____,
Example # 2. (d)
Look for a pattern and predict the next two
numbers or letters.
34 ____
55
1, 1, 2, 3, 5, 8, 13, 21, ____,
Example # 2. (e)
Look for a pattern and predict the next two
numbers or letters.
E ____
N
O, T, T, F, F, S, S, ____,
Example # 2. (f)
Look for a pattern and predict the next two
numbers or letters.
S ____
N
J , M , M , J , ____,
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.
∠1 ∠ 2
Hannah concludes that ______________.
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.
AB CD and AB ^ XY
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 $30 - 5 = $25
3 x $9 = $27
$27 - 2 (bellhop) = $25
Patterns in Mathematics
1x8+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
Assignment
Written Exercises on page 108
DO with CALCULATOR NOW: 15, 16, 17
How can you apply parallel
lines (planes) to make
deductions?
Patterns in Mathematics
1 x 9 + 2 = 11
Page 108 #15
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
Patterns in Mathematics
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
Page 108 #16
Patterns in Mathematics
92 = 81
992 = 9801
9992 = 998001
99992 = 99980001
999992 = 9999800001
9999992 = 999998000001
99999992 = 99999980000001
999999992 = 9999999800000001
9999999992 = 999999998000000001
Page 108 #17
Assignment
Written Exercises on page 107
GRADED: 1 to 13 all numbers
How can you apply parallel
lines (planes) to make
deductions?
Class Assignment:
Chapter Review on pages 111 & 112
5 to 10 all numbers, 12, 14,
17 to 21 all numbers
Prepare for a test on
Chapter 3: Parallel Lines and Planes
How can you apply parallel
lines (planes) to make
deductions?
Class Assignment:
Chapter Test on pages 112
1 to 6 all numbers
9 to 11 all numbers
Prepare for a test on
Chapter 3: Parallel Lines and Planes
How can you apply parallel
lines (planes) to make
deductions?