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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
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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
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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
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−−− −−
4. If DE ⊥ EF, then ∠ DEF is a right angle.
6
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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.
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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.
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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.
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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.
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