Transcript 2.1 Inductive Reasoning

Using
Inductive
Reasoning
to
Using
Inductive
Reasoning
to
2-1
2-1 Make
Conjectures
Make
Conjectures
Section 2.1
Holt
Geometry
Holt
McDougal
Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Warm Up
Complete each sentence.
1.
2.
?
?
points are points that lie on the same line.
points are points that lie in the same plane.
3. The sum of the measures of two
Holt McDougal Geometry
?
angles is 90°.
Using Inductive Reasoning to
2-1 Make Conjectures
• Inductive Reasoning is when one looks at
several facts and then makes an educated
guess on these facts.
• That educated guess is called a
conjecture. (Which may or may not be
true.)
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Find the next item in the pattern.
January, March, May, ...
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Find the next item in the pattern.
7, 14, 21, 28, …
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Find the next item in the pattern.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Find the next item in the pattern 0.4, 0.04, 0.004, …
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Complete the conjecture.
The sum of two positive numbers is
Holt McDougal Geometry
? .
Using Inductive Reasoning to
2-1 Make Conjectures
Complete the conjecture.
The number of lines formed by 4 points, no
three of which are collinear, is ? .
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Complete the conjecture.
The product of two odd numbers is
Holt McDougal Geometry
? .
Using Inductive Reasoning to
2-1 Make Conjectures
To show that a conjecture is always true, you must
prove it.
To show that a conjecture is false, you have to find
only one example in which the conjecture is not true.
This case is called a counterexample.
A counterexample can be a drawing, a statement, or a
number.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a
counterexample.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Show that the conjecture is false by finding a
counterexample.
For every integer n, n3 is positive.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Show that the conjecture is false by finding a
counterexample.
Two complementary angles are not congruent.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Show that the conjecture is false by finding a
counterexample.
The monthly high temperature in Abilene is
never below 90°F for two months in a row.
Monthly High Temperatures (ºF) in Abilene, Texas
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug Sep
88
89
97
99
107
109
110
107 106 103
Holt McDougal Geometry
Oct Nov Dec
92
89
Using Inductive Reasoning to
2-1 Make Conjectures
Show that the conjecture is false by finding a
counterexample.
For any real number x, x2 ≥ x.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Show that the conjecture is false by finding a
counterexample.
Supplementary angles are adjacent.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Determine if each conjecture is true or false based on
the information. If false, give a counter-example.
1.Given: AB, BC and CD
Conjecture: A, B, C and D are collinear.
2. Given: ∠1 and ∠2 form a linear pair.
Conjecture: m ∠1 + m ∠2 = 180
3.Given: ∠1 and ∠2 are supplementary.
Conjecture: ∠1 = ∠2
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
4. Given: P, Q and R are collinear.
Conjecture: Q is in between P and R.
5. Given: ∠1 and ∠2 are supplementary
Given: ∠1 and ∠3 are supplementary
Conjecture: ∠2 = ∠ 3
6. Given: points D, E, F, G
Conjecture: D, E, F, G are noncollinear
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
7. Given: Collinear points X, Y and Z
Given: Z is between X and Y
Conjecture: XY + YZ = XZ
8. Given: Noncollinear points L, M, N
Conjecture: LM, MN and LN form a triangle
9. Given: PQRS is a rectangle
Conjecture: PQ = RS and QR = SP
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Lesson Quiz
Find the next item in each pattern.
1. 0.7, 0.07, 0.007, … 2.
Determine if each conjecture is true. If false,
give a counterexample.
3. The quotient of two negative numbers is a positive
number.
4. Every prime number is odd.
5. Two supplementary angles are not congruent.
6. The square of an odd integer is odd.
Holt McDougal Geometry