1. Using Inductive Reasoning to Make Conjectures 2-1
Section 2.1
Holt McDougal Geometry
Holt Geometry

2. Complete each sentence.
Warm Up
Complete each sentence.
1. ? points are points that lie on the same line.
2. ? points are points that lie in the same plane.
3. The sum of the measures of two ? angles is 90°.

3. 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.)

4. Find the next item in the pattern.
January, March, May, ...

5. Find the next item in the pattern.
7, 14, 21, 28, …

6. Find the next item in the pattern.

7. Find the next item in the pattern 0.4, 0.04, 0.004, …

8. Complete the conjecture.
The sum of two positive numbers is ? .

9. Complete the conjecture.
The number of lines formed by 4 points, no three of which are collinear, is ? .

10. Complete the conjecture.
The product of two odd numbers is ? .

11. 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.

12. Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a counterexample.

13. Show that the conjecture is false by finding a counterexample.
For every integer n, n3 is positive.

14. Show that the conjecture is false by finding a counterexample.
Two complementary angles are not congruent.

15. Monthly High Temperatures (ºF) in Abilene, Texas
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
Oct
Nov
Dec 88 89 97 99
107
109
110
106
103 92

16. Show that the conjecture is false by finding a counterexample.
For any real number x, x2 ≥ x.

17. Show that the conjecture is false by finding a counterexample.
Supplementary angles are adjacent.

18. Determine if each conjecture is true or false based on the information
Determine if each conjecture is true or false based on the information. If false, give a counter-example.
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
Given: ∠1 and ∠2 are supplementary.
Conjecture: ∠1 = ∠2

19. 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

20. 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

21. 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.