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Lesson 2-1 (pg 74)
Assgn 2-1: #1-21,25-31,35-39
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
8, 3, 2, 7, . . .
Conjecture
Each term is 5 more than the
previous term.
Next Two Items
7  5  12
12  5  17
The measure of each angle is
half the measure of the
previous angle.
Notes:
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Find the next item in each pattern.
1.
1 1 3
, , , 1, . . .
4 2 4
2. 100, 81, 64, 49, . . .
________________________________________
3.
________________________________________
4.
________________________________________
________________________________________
Complete each conjecture.
5. If the side length of a square is doubled, the perimeter of the square
is __________________________________ .
6. The number of nonoverlapping angles formed by n lines intersecting in a point
is __________________________________ .
Use the figure to complete the conjecture in Exercise 7.
7. The perimeter of a figure that has n of these triangles
is __________________________________ .
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Lesson 2-1 cont
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)
00
true
2
2  4(2)
2  8
false
n  2 is a counterexample, so the conjecture is false.
Notes:
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Holt McDougal Geometry
Name _______________________________________ Date __________________ Class __________________
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.
________________________________________________________________________________________
9. Points A, G, and N are collinear. If AG  7 inches and GN  5 inches, then
AN  12 inches.
________________________________________________________________________________________
10. For any real numbers x and y, if x  y, then x 2  y 2.
________________________________________________________________________________________
11. The total number of angles in the figure is 3.
________________________________________
________________________________________
12. If two angles are acute, then the sum of their measures equals the
measure of an obtuse angle.
________________________________________
________________________________________
Determine whether each conjecture is true. If not, write or draw a counterexample.
13. Points Q and R are collinear.
________________________________________
14. If J is between H and K, then HJ  JK.
________________________________________
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Answers for the chapter Geometric Reasoning
USING INDUCTIVE REASONING TO
MAKE CONJECTURES
Practice A
1. 10
2. W
3. summer
3. The pattern is the letters of the alphabet
that are made only from straight
segments; K, L.
4. First rotate the figure 180. Then reflect
the figure across a vertical line. Repeat.
5. true
6. false
4. inductive reasoning
5. true
6. even
Sample answer:
7. n
8. The number of rings in a tree is the
same as the tree’s age.
9. 82 rings
7. true
8. false
Possible answers: n  1, n  1
10. false
11. Possible answers: zero, any negative
number
9. Sum of angle measures  [180(n  2)]
10.128.6°; 140°; 150°
Reteach
1
1. 1
4
12. Possible answer:
Practice B
1. 36
2.
3. Arkansas
4. north
5. positive
6. n  3
2. 36
3.
4.
7. Possible answers: zero, any negative
number
5. doubled
6. 2n
8.
8. Possible answer: If the lines are
parallel, then they do not intersect.
9. One-third of the bills were counterfeit.
10. Each item, starting with the third, is the
product of the two preceding items;
256, 8192.
11. The dot skips over one vertex in a
clockwise direction.
Practice C
1. The pattern is the cubes of the negative
integers; 125, 216.
2. Each item describes the item before it
(one, one one, two ones, . . .); 312211,
13112221.
7. n  2
9. Possible answer: If point N is between
points A and G, then AN  2 inches.
10. Sample answer: If x  0 and y  1,
then x 2  y 2.
11. Sample answer: ABD, DBE, EBC,
ABE, DBC
12. Sample answer: m1  25°, m2  20°
13. true
14.
Challenge
1. 1, 6, 15, 20, 15, 6, 1
2.Each row has 1 as the first and last
number. Each of the other numbers is
found by adding the two numbers that
appear just above it.
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