Inequalities

And

Indirect Proofs

In Geometry

Honors

Table of Contents

DAY 1: (Ch 15-1) SWBAT: Write simple Inequality Proofs Pgs: 1-7 HW: 8-10 in packet

DAY 2: (Ch. 15-2) SWBAT: Write Inequality Proofs using Inequalities in Triangles Pgs: 11-15

HW: 16-18 in packet

DAY 3: (Ch. 15-1 to 15-2) SWBAT: Practice Writing Inequality Proofs Pgs: 19-22

HW: Finish the section

DAY 4: (5-1) SWBAT: Write Indirect Proofs

Pgs: 23-26

HW: 27-29 in packet

DAY 5: (5-1) SWBAT: More Practice Writing Indirect Proofs

Pgs: 30-32

HW: Finish the section

DAY 6: Review

Day 7: Test

1

Inequality Postulates and Theorems

Postulate #1: A whole is greater than each of its parts.

2

Transitive Property of Inequality

Postulate #2:

Substitution Postulate of Inequality

Postulate #3:

Model Problems

3

Addition Postulate of Inequality

Postulate #4:

Postulate #5:

Model Problems

4

Subtraction Postulate of Inequality:

Postulate #6:

Model Problems

5

Multiplication Postulate of Inequality:

Postulate #7:

Division Postulate of Inequality:

Postulate #8:

6

Model Problems

7

SUMMARY

Method 1: Method 2:

(Addition Postulate of Inequality) (1, 2)

8

Homework

6.

7.

9

10

E is the midpoint of , F is the midpoint of

11

Day 2 - Inequalities involving Triangles

Warm – Up

Inequality involving the lengths of the sides of a triangle

Postulate #9:

Inequalities involving the exterior angle of a triangle

12

Postulate #10:

Postulate #11:

Postulate #12:

Model Problem

1.

13

2.

Postulate #13

14

Postulate #14

Model Problems

3.

15

4.

SUMMARY

1. fdfd

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2.

______________________________________________________

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16

Homework

17

18

19

Day 3 - Practice With inequality Proofs

Warm - Up

20

1. Given:

2. Given: 1 2

Prove: RM > MS

21

T

R SK M

21

3.

4.

22

5.

6.

23

Day 4 - Indirect Proofs – Proof By Contradiction

When trying to prove a statement is true, it may be beneficial to ask yourself,

"What if this statement was not true?" and examine what happens. This is the

premise of the Indirect Proof or Proof by Contradiction.

Indirect Proof: Assume what you need to prove is false, and then show that something

contradictory (absurd) happens.

Steps in an Indirect Proof:

Assume that the opposite of what you are trying to prove is true.

From this assumption, see what conclusions can be drawn. These

conclusions must be based upon the assumption and the use of valid

statements.

Search for a conclusion that you know is false because it contradicts

given or known information. Oftentimes you will be contradicting a

piece of GIVEN information.

Since your assumption leads to a false conclusion, the assumption must

be false.

If the assumption (which is the opposite of what you are trying to prove)

is false, then you will know that what you are trying to prove must be

true.

24

Example #1

Given: m A = 50 and mB = 70

Prove: A and B are not complementary

Example #2

Given :

Prove: and do not bisect each other.

25

3. Given: ΔABC is scalene

BD bisects ABC.

Prove: BD is not perpendicular to AC

4.

A

B

C D

1 2

3 4

26

SUMMARY

Assumption leading to a Contradiction

9. B is not to CED 9. Contradiction (7, 8)

27

Homework

28

4. Given: is scalene

Prove:

5. Given: BE is the median of AC ,

Prove: ΔABC is not Isosceles

CBEABE

29

6.

30

Day 5 - More Indirect Proofs

Prove each of the following indirectly.

1. Given:

BAC DAC

Prove:

2. Given: l m

Prove: 1 2

A

BDC

m

l

t

21

31

3.

4. Prove that if ABC is isosceles with base and if P is a point on that is not

the midpoint, and then does not bisect BAC.

32

5. Given: 1 2

ABCD is not a parallelogram

Prove: 3 4

6. Given: O

is not an altitude

Prove: does not bisect AOC

O

CA

B