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Warm-up

- Get all papers passed back
- Get out a new sheet for ch. 6 warm-ups
- Solve ..
- x 2 gt 5
- x 9 lt 7
- 2x gt 10
- 10 lt -5x

Write down one rule that is used when dealing

with inequalities. Think back to algebra.

Chapter 6Inequalities in Geometry

6-1 Inequalities

- Objectives
- Apply properties of inequality to positive

numbers, lengths of segments, and measures of

angles - State and use the Exterior
- Angle Inequality Theorem.

Law of Trichotomy

- The "Law of Trichotomy" says that only one of the

following is true

- Alex Has Less Money Than Billy or
- Alex Has the same amount of money that Billy has

or - Alex Has More Money Than Billy

Makes Sense Right !

Equalities vs Inequalities

- To this point we have dealt with congruent
- Segments
- Angles
- Triangles
- Polygons

Equalities vs Inequalities

- In this chapter we will work with
- segments having unequal lengths
- Angles having unequal measures

The 4 Inequalities

Symbol Words

gt greater than

lt less than

greater than or equal to

less than or equal to

The symbol "points at" the smaller value

Does a lt b mean the same as b gt a?

A review of some properties of inequalities (p.

204)

- When you use any of these in a proof, you can

write as your reason, A property of Inequality

1. If a lt b, then a c lt b c

If a lt b, then a c lt b c

Example

- Alex has less coins than Billy.
- If both Alex and Billy get 3 more coins each,

Alex will still have less coins than Billy.

Likewise

- If a lt b, then a - c lt b - c
- If a gt b, then a c gt b c, and
- If a gt b, then a - c gt b - c

So adding (or subtracting) the same value to both

a and b will not change the inequality

2. If a lt b, and c is positive, then ac lt bc

3. If a lt b, and c is negative, then ac gt bc

(inequality swaps over!)

This is true for division also !

- If a lt b, and c is positive, then a lt b c

cIf a lt b, and c is negative, then a gt b - c c
- Who can provide an example of this inequality?

4. If a lt b and b lt c, then a lt c

If a lt b and b lt c, then a lt c

Example

- 1.) If Alex is younger than Billy and
- 2.) Billy is younger than Carol,
- Then Alex must be younger than Carol also!

5. If a b c and b c are gt 0, then a gt b

and a gt c

Why ?

The Exterior Angle Inequality Theorem

- The measure of an exterior angle of a triangle is

greater than the measure of either remote

interior angle.

- Remember the exterior angle theorem? Based on

the diagram, mL4 _____ ______

2

m ? 4 gt m ? 1

m ? 4 gt m ? 2

1

4

3

Remote time

If a and b are real numbers and a lt b, which one

of the following must be true?

- A. -a lt -b
- B. -a gt -b
- C. a lt -b
- -a gt b
- I dont know

True or False

- If XY YZ 15, then XY gt YZ
- If m ? A m ? B m ? C, then m ? B gt m ? C
- If m ? H m ? J m ? K, then m ? K gt m ? H
- If 10 y 2, then y gt 10

White Board Practice

- Given RS lt ST STlt RT
- Conclusion RS ___ RT

R

S

T

White Board Practice

- Given RS lt ST STlt RT
- Conclusion RS lt RT

R

S

T

White Board Practice

- Given m ? PQU m ?PQT m ?TQU
- Conclusion m ? PQU ____ m ?TQU
- m ? PQU ____ m ?PQT

U

T

P

R

Q

White Board Practice

- Given m ? PQU m ?PQT m ?TQU
- Conclusion m ? PQU gt m ?TQU
- m ? PQU gt m ?PQT

U

T

P

R

Q

6-2 Inverses and Contrapositives

- State the contrapositives and inverse of an

if-then statement. - Understand the relationship between logically

equivalent statements. - Draw correct conclusions from given statements.

Warm up

- Identify the hypothesis and the conclusion of

each statements. Then write the converse of each. - If Maria gets home from the football game late,

then she will be grounded. - If Maria is grounded, then she got home from the

football game late. - If Mike eats three happy meals, then he will have

a major stomach ache. - If Mike has a major stomach ache, then he ate

three happy meals.

Venn Diagrams

ALL IF/THEN STATEMENTS CAN BE SHOWN USING A VENN

DIAGRAM.

THEN

- Geographical boundaries are created
- Take the statement and put the hypothesis and

conclusion with in these boundaries - Example Think of a fugitive and his

whereabouts City/State

IF

Venn Diagrams

If Maria gets home from the football game late,

then she will be grounded.

Venn Diagrams

If Mike eats three happy meals, then he will have

a major stomach ache.

Venn Diagrams

If we have a true conditional statement, then we

know that the hypothesis leads to the conclusion.

Summary of If-Then Statements

Statement How to remember Symbols Example

Conditional The Original If p, then q If the fugitive is in LA, then he is in CA.

Converse Reverse of the original If q, then p If the fugitive is in CA, then he is in LA.

Inverse Opposite of the original If not p, then not q If the fugitive is NOT in LA, then he is NOT in CA.

Contrapositive Opposite the reverse If not q, then not p If the fugitive is NOT in CA, then he is NOT in LA.

Logically Equivalent

- Conditional
- Contrapositve

- Inverse
- Converse

THESE STATEMENTS ARE EITHER BOTH TRUE OR BOTH

FALSE!!!

Conditional / ContrapostiveLogically Equivalent

If Mike eats three happy meals, then he will have

a major stomach ache.

If Mike did not have a major stomach ache, then

he did not eat three happy meals.

Its a funny thing

- This part of geometry is called LOGIC, however,

if you try and think logically you will usually

get the question wrong. - Let me show you

Venn Diagrams

What do the other colored circles represent?

THEN

IF

Arent there other reasons why Maria might get

grounded?

Then she is grounded

Late from football game

Arent there other reasons why Mike might get a

stomach ache?

Has a major stomach ache

Eats three happy meals

Example 1

- If it is snowing, then the game is canceled.
- What can you conclude if I say, the game was

cancelled?

Example 1

Nothing !

- If it is snowing, then the game is canceled.
- What can you conclude if I say, the game was

cancelled?

There are other reasons that the game would be

cancelled

Game cancelled

D

B

Snowing

A

C

- All you can conclude it that it MIGHT be snowing

and that isnt much of a conclusion.

Lets try again

- Remember dont think logically. Think about

where to put the star in the venn diagram.

Example 2

- If you are in Coach Gosss class, then you have

homework every night. - a) What can you conclude if I tell you Jim has

homework every night?

Jim might be in Coach Gosss classNo Conclusion

Homework every night

D

A

Coach Gosss class

B

C

Example 3

- If you are in coach Gosss class, then you have

homework every night. - b) What can you conclude if I tell you Rob is in

my 2nd period?

Rob has homework every night

Homework every night

D

A

Coach Gosss class

B

C

Example 4

- If you are in Coach Gosss class, then you have

homework every night. - b) What can you conclude if I tell you Bill has

Mr. Brady?

Bill might have homework every nightNo conclusion

E

Homework every night

D

A

Coach Gosss class

B

C

Example 5

- If you are in coach Gosss class, then you have

homework every night. - d) What can you conclude if I tell you Matt

never has homework?

Matt is not in my class

E

Homework every night

D

A

Coach Gosss class

B

C

NOTES - EXAMPLE

- If the sun shines, then we go on a picnic.
- What can you conclude if
- a) We go on a picnic
- b) The sun shines
- c) It is raining
- d) We do not go on a picnic

- a) We go on a picnic no conclusion
- b) The sun shines We go on a picnic
- c) It is raining no conclusion
- We do not go on a picnic
- The sun is not shining

We go on a picnic

Sun shines

White Board Practice

- All runners are athletes.
- What can you conclude if
- a) Leroy is a runner
- b) Lucy is not an athlete
- c) Linda is an athlete
- d) Larry is not a runner

First the statement MUST be in the form

if________, then_______

- All runners are athletes
- If you are a runner, then you are an athlete

White Board Practice

- All runners are athletes.
- What can you conclude if
- a) Leroy is a runner
- b) Lucy is not an athlete
- c) Linda is an athlete
- d) Larry is not a runner

a) Leroy is a runner He is an

athlete b) Lucy is not an athlete She is not

a runner c) Linda is an athlete

no conclusion d) Larry is not a runner

no conclusion

You are an athlete

Runner

Warm - up

- Write a statement that is logically equivalent to

the following - If it is Sunday, then John takes the trash out.
- Write in if-then form
- All marathoners have stamina
- What conclusions can you come to
- a. Nick is a marathoner
- b. Heidi has stamina
- c. Mimi does not have stamina
- d. Arlo is not a marathoner

Warm - up

- Write a statement that is logically equivalent to

the following - If it is Sunday, then John takes the trash out.
- If Bob cant swim, then he will no enter the

race

6-3 Indirect Proof

- Objectives
- Write indirect proofs in paragraph form

- After walking home, Sue enters the house carrying

a dry umbrella. - We can conclude that it is not raining outside.

Why?

- Because if it HAD been raining, then her umbrella

would be wet. - The umbrella is not wet.
- Therefore, it is not raining.

How do you feel about proofs?

- I dont like them at all
- I dont mind doing them
- I havent learned all of the definitions/postulate

s/ and theorems, so they are still hard for me to

do. - I love doing proofs
- Im getting better at doing proofs

UUGGGHHH more proofs

- Up until now the proofs that you have written

have been direct proofs. - Sometimes it is IMPOSSIBLE to find a direct proof.

Indirect Proof

- Are used when you cant use a direct proof.
- BUT, people use indirect proofs everyday to

figure out things in their everyday lives. - 3 steps EVERYTIME (p. 214 purple box)
- GIVEN Dry Umbrella
- Prove It wasnt raining

Step 1

- Assume temporarily that. (the opposite of the

conclusion ). - You want to believe that the opposite of the

conclusion is true (the prove statement). - Assume temporarily that it was raining outside.

Step 2

- Using the given information or anything else that

you already know for sure..(like postulates,

theorems, and definitions) - try and show that the temporary assumption that

you made cant be true. - You are looking for a contradiction to the GIVEN

information. - This contradicts the given information.
- Use pictures and write in a paragraph.
- If it HAD been raining, then her umbrella would

be wet. This contradicts the given information

that the umbrella was dry.

Hudsons Dilemma

- Hudson and his girlfriend drove to the river at

Parker, AZ to meet up with her Parents. - They left Redondo Beach at noon and arrived in

Parker at 6pm. - The total trip was 510 miles
- When they arrived, his girlfriends mom

exclaimed, Wow you got here fast, you must have

been speeding!! - Hudson replied, Oh no, we went the speed limit

(65mph) the whole time - HOW DID HIS GIRLFRIENDS MOM FIGURE OUT HE WAS

LYING????

Step 3

- Point out that the temporary assumption must be

false, and that the conclusion must then be true. - My temporary assumption is false and

(the original conclusion must be true). Restate

the original conclusion. - Therefore, my temporary assumption is false and

it was not raining outside.

Given Hudson drove 510 miles to the river in 6

hours.Prove Hudson exceeded the 65 mph speed

limit while driving.

- Step 1 Assume temporarily that Hudson did not

exceed the 65 mph - Step 2 Then the minimum time it would take

Hudson to get to the river is 510/65 7.8 hours.

This is a contradiction to the given information

that he got there in 6 hours. - Step 3 My temporary assumption is false and

Hudson exceeded the 65 mph speed limit while

driving.

Given Hudson drove 510 miles to the river in 6

hours.Prove Hudson exceeded the 65 mph speed

limit while driving.

- Assume temporarily that Hudson did not exceed

the 65 mph. Then the minimum time it would take

Hudson to get to the river is 510/65 7.8 hours.

This is a contradiction to the given information

that he got there in 6 hours. My temporary

assumption is false and Hudson exceeded the 65

mph speed limit while driving.

Example 3

- Given Trapezoid PQRS with bases PQ and SR
- Prove PQ ?SR

Given Trapezoid PQRS with bases PQ and

SRProve PQ ?SR

- Step 1 Assume temporarily PQ SR

Given Trapezoid PQRS with bases PQ and

SRProve PQ ?SR

- Step 1 Assume temporarily PQ SR
- Step 2 Since PQRS is a trapezoid and PQ and SR

are the bases, I know by the definition of a

trapezoid, that PQ SR. If PQ SR and PQ

SR, then PQRS is a parallelogram because If one

pair of opposite sides of a quadrilateral are

both ? and , then the quadrilateral is a

parallelogram. This contradicts the given

information that PQRS is a trapezoid, because a

quadrilateral cant be a trapezoid AND a

parallelogram.

Given Trapezoid PQRS with bases PQ and

SRProve PQ ?SR

- Step 1 Assume temporarily PQ SR
- Step 2 Since PQRS is a trapezoid and PQ and SR

are the bases, I know by the definition of a

trapezoid, that PQ SR. If PQ SR and PQ

SR, then PQRS is a parallelogram because If one

pair of opposite sides of a quadrilateral are

both ? and , then the quadrilateral is a

parallelogram. This contradicts the given

information that PQRS is a trapezoid, because a

quadrilateral cant be a trapezoid AND a

parallelogram. - Step 3 My temporary assumption is false and PQ

?SR

Given Trapezoid PQRS with bases PQ and

SRProve PQ ?SR

- Assume temporarily PQ SR. Since PQRS is a

trapezoid and PQ and SR are the bases, I know by

the definition of a trapezoid, that PQ SR. If

PQ SR and PQ SR, then PQRS is a parallelogram

because If one pair of opposite sides of a

quadrilateral are both ? and , then the

quadrilateral is a parallelogram. This

contradicts the given information that PQRS is a

trapezoid, because a quadrilateral cant be a

trapezoid AND a parallelogram. My temporary

assumption is false and PQ ?SR

White board practice

- Write an indirect proof in paragraph form
- Given m ? X ? m ? Y
- Prove ? X and ? Y are not both right angles

Given m ? X ? m ? YProve ? X and ? Y are not

both right angles

- Assume temporarily that ? X and ? Y are both

right angles. I know that m ? X 90 and m ? Y

90, because of the definition of a right angle.

If the m ? X 90 and m ? Y 90, then by

substitution, m ? X m ? Y. This is a

contradiction to the given information that m ? X

? m ? Y. My teomporary assumption is false and ?

X and ? Y are not both right angles

White board practice

- Write an indirect proof in paragraph form
- Given ? XYZW m ? X 80º
- Prove ? XYZW is not a rectangle

Given ? XYZW m ? X 80ºProve ? XYZW is not

a rectangle

- Assume temporarily that ? XYZW is a rectangle.

Then ? XYZW have four right angles because this

is the definition of a rectangle. This

contradicts the given information that m ? X

80º. My temporary assumption is false and ?

XYZW is not a rectangle.

Quiz Review 6.1 6.3

- Section 6.1
- First Warm-up
- Hw 50 s 1 4
- Section 6.2
- Writing the contrapostive and inverse of a

statement (Is it true or false?) - Warm-up 2 also, hw 51 s 5, 7, 9
- Understand what statements are logically

equivalent - Pg. 208 pink box
- What can you conclude from the statement?
- Hw 52 s 12 18
- Section 6.3
- Know the 3 steps to writing an indirect proof
- Hw 53 s 1 5 Study worksheet

- If y gt x, then x lt y.
- If g lt t and g gt r , then t gt r.
- If c lt r and c f , then r gt f.
- If you are a freshman, then you are in high

school. - Write the inverse. T/F?
- If you are not a freshman, then you are not in

high school. F - Write the contrapostive T/F?
- If you are not in high school, then you are not

a freshman. T

- All parallelograms are quadrilaterals
- ABCD is a quad
- GHTS is a parallelogram
- TRIS is not a parallelogram
- PRS is a triangle
- If 2 lines are CBT, then corr. angles are

congruent. - Write the first sentence if trying to prove this

with an indirect proof.

6-4 Inequalities for One Triangle

- Objectives
- State and apply the inequality theorems and

corollaries for one triangle.

Remember the Isosceles Triangle Theorem

- If two sides of a triangle are congruent then the

angles opposite those sides are congruent.

So what do you think we can say if the two sides

are not equal?

B

AB gt BC

Partners Create a hypothesis stating the

relationship between the side lengths of a

triangle in comparison to the angles opposite

those sides. Be able to explain it as well.

A

C

Theorem

- If one side of a triangle is longer than a second

side, then the angle opposite the first side is

greater than the angle opposite the second side.

12

15

this angle is larger than the other angle

White Board Practice

- Name the largest angle and the smallest angle of

the triangle.

H

10

6

I

J

8

Theorem

- If one angle of a triangle is larger than a

second angle, then the side opposite the first

angle is longer than the side opposite the second

angle.

B

What do you think we can conclude if this angle

measurement is greater than the other angle

measurement?

A

C

White Board Practice

- Name the largest side and the shortest side of

the triangle.

T

105

46

R

S

Corollary 1

- The perpendicular segment from a point to a line

in the shortest segment from the point to the

line.

Which line looks shorter?

The green and black line look like they are what?

This is why the legs of a right triangle are

always shorter than the hypotenuse.

Corollary 2

- The perpendicular segment from a point to a plane

in the shortest segment from the point to the

plane.

- Who knows the old saying The shortest distance

between two points is.

A STRAIGHT LINE!!

SF

VEGAS

LA

The Triangle Inequality Theorem

- The sum of the lengths of any two sides of a

triangle is greater than the length of the third

side.

a b gt c a c gt b b c gt a Always label your

sides a, b, c and then just follow the rules

of the theorem.

a

c

b

White Board Practice

- The length of two sides of a triangle are 8 and

13. Then, the length of the third side must be

greater than_______ but less than _______.

13

13 8 gt c 13 c gt 8 8 c gt 13

c

8

White Board Practice

- The length of two sides of a triangle are 8 and

13. Then, the length of the third side must be

greater than_______ but less than _______.

13 8 gt c 13 c gt 8

8 c gt 13 21 gt c c gt -5

c gt 5 c lt 21

White Board Practice

- The length of two sides of a triangle are 8 and

13. Then, the length of the third side must be

greater than 5 but less than 21 .

13 8 gt c 13 c gt 8

8 c gt 13 21 gt c c gt -5

c gt 5 c lt 21

The 3rd side must be greater than the difference

of the 2 s The 3rd side must be less than the

sum of the 2 s

White Board Practice

- Is it possible for a triangle to have sides with

lengths 16, 11, 5 ?

NO

16 11 gt 5 16 5 gt 11

5 11 gt 16 27 gt 5 21 gt 11

16 gt 16

USE THE COVER UP METHOD

White Board Practice

- Is it possible for a triangle to have sides with

the lengths indicated? - Yes No

6, 8, 10

Yes

3, 4, 8

No

4, 6, 2

No

6, 6, 5

Yes

Warm Up

- Write an indirect proof in paragraph form
- Given ? XYZW XY 10 YZ 12
- Prove ? XYZW is not a rhombus

6-5 Inequalities for Two Triangles

- Objectives
- State and apply the inequality theorems for two

triangles

Remember SAS and SSS

- What do each of these mean?
- We are going to use the general basis of these

theorems and apply them to triangles with unequal

side lengths and angle measures

Triangle Experiment

- Supplies 3 pencils or pens
- Step 1 Place two pencils together to form an

acute angle (the two pencils are the sides of a

triangle, without the third side) - Step 2 With the open end of your triangle on

your paper. Draw a line connecting the open ends

to form the 3rd side of your triangle.

Triangle Experiment

- Step 3 Now, INCREASE the size of the angle

created by the two pencils. Draw another line

connecting the open ends.

Triangle Experiment

- Essentially you have created 2 different

triangles. These triangles have 2 sides

congruent to each other. What do you notice

about the relationship between the included angle

measurement and the length of the 3rd side?

Write it down

Make a People Triangle

- Step 1 Measure students' heights and identify

two students who are identical in height to two

other students. - Step 2 Have two of the students lie on the

floor, their feet touching at an angle, to form

two sides of a triangle, and measure the distance

between the students' heads. - Step 3 Do the same thing with the second pair

of students. smaller angle.

What did we find?

- The distance between the heads of the students

who made the bigger angle was greater than the

distance between the heads of the students who

made the smaller angle.

SAS Inequality Theorem

E

B

C

D

F

A

SSS Inequality Theorem

B

E

C

D

A

F

White Board Practice

- Given D is the midpoint of AC m ? 1lt m ? 2
- What can you deduce?

B

1

2

A

C

D

Complete with lt, , or gt

- m ? 1_ gt _ m ? 2

4

4

1

2

3

4

Whiteboard Practice

- Page 230
- 1
- LCAB gt LFDE SSS ineq.
- 3
- RT gt TS SAS ineq.
- 7
- L1 gt L2 SSS ineq.

Test Review

- Section 6.1
- Theorem 6-1 exterior angle inequality (p 204)
- Self test 1 s 1 4 and Quiz s

1 3 - Section 6.2
- Writing a statement in if-then form
- Also writing Converse, contrapostive and inverse

of a statement (Is it true or false?) - Read pg. 208 / hw 51 s 5, 7, 9
- What can you conclude from the statement? (hint

draw venn diagram!!) - Hw 52 s 12 18 / quiz 7
- Section 6.3
- Know the 3 steps to writing an indirect proof
- Hw 53 s 1 5 Study worksheet and

notes - Know each specific phrase that goes with each

step!

Test Review

- Section 6.4
- The sum of 2 sides of a triangle is always

greater than the third side. (p. 222 1 4) - Theorem 6-2 and 6-3
- P. 221 1 6
- pg. 223 16
- Section 6.5
- 6.5 worksheet
- P. 231 2