6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are...

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6.1 Inequalities and Indirect Proofs

Transcript of 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are...

Page 1: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

6.1 Inequalities and Indirect Proofs

Page 2: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________ in our proofs.

Now we will deal with _________ sides and angles. We will be using properties of ______________.

Page 3: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Property of Inequality

(All of these)

Page 4: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Exterior Angle Theorem: ____________

__________________________________

__________________________________

__________________________________

1

2

3

Page 5: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

: ;

Prove :

Given AC BC CE CD

AE BD

A

B

C

D

E

Page 6: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

: 1 is and exterior angle

Prove : 1 ; 1

Given

D E

F

D

E

1

Page 7: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

: 2> 1

Prove : 2 4

Given

1

3

2

4

Page 8: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

6.3 Indirect Proof

Page 9: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

We will be using __________________

_____________________________. Lets look at an example.

Lets suppose after walking home, Joe enters the house carrying a dry umbrella. You can conclude that it is not raining outside. Why?

________________________________

________________________________

Page 10: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Steps for solving an Indirect Proof:

•___________________________________________________________________

•___________________________________________________________________

•_______________________________________________________________________________________________________________________________________

Page 11: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Lets try one, these will be written as a paragragh.

: is a trapezoid

Prove: PQ SR

Given SRQP

S R

P Q

__________________________________________

__________________________________________

__________________________________________

__________________________________________

__________________________________________

__________________________________________

Page 12: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

6.4

Inequalities

Page 13: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Inequalities for one Triangle

Theorem 6.2: __________________________

________________________________________

________________________________________

________________________________________

Theorem 6.3: __________________________

________________________________________

________________________________________

________________________________________

Page 14: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

A

B

C

10 12

18

List angles from Largest to smallest:

____ ____ ____

61

60 59

A

BC

List sides from Largest to smallest:

____ ____ ____

Page 15: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Corollary 1: ______________________________

________________________________________

________________________________________

Theorem 6.4: ___________________________

________________________________________

________________________________________

Page 16: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

When given 2 sides of a triangle you can find a range that the third side will be between.

•_____________________________________

______________________________________

______________________________________

10,12, ________

25,26, ________

, 2, _______y y

Page 17: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Find out if each is a triangle, given the sides:

1. 6,8,20

2. 2.5, 5, 4.1

3. 3, 4, 5

4. 6, 4, 2

5. 6, 5, 6

Page 18: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

80

x x-10

B

AC

Which side is the longest?

Page 19: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

A

B

C

D61

59

60

61

60

59

____ ____ ____ ____ ____

Page 20: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

6.5 Inequalities for 2 Triangles

Page 21: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Theorem 6-5: SAS Inequality

______________________________________________________________________________________________________________________________________________________________________________________________________________________________

55 34

A B

Page 22: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Theorem 6-5: SSS Inequality

______________________________________________________________________________________________________________________________________________________________________________________________________________________________

55 34

A B

Page 23: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Example:

What can you deduce?

: ; 1 2Given RS RT

1

2R

T

V

S

•____________

•___________________________________________________

Page 24: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Example:

What can you deduce?

: Marked on DrawingGiven

R

T

V

S

•______________

•______________

Y

X

12

14

Page 25: 6.1 Inequalities and Indirect Proofs. Up until now we have dealt with sides and angles that are ___________ and have used the properties of _________.

Example:

What can you deduce?

: Marked on DrawingGiven

5

10

•______________

•______________10

1

6

2