Warm-up

• Get all papers passed back

• Get out a new sheet for ch. 6 warm-ups

• Solve ..– x + 2 > 5– x – 9 < 7– 2x > 10 – 10 < -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

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

> greater than

< less than

≥ greater than or equal to

≤ less than or equal to

The symbol "points at" the smaller value

Does a < b mean the same as b > 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 < b, then a + c < b + c

If a < b, then a + c < b + c

Alex has less coins than Billy.

• If both Alex and Billy get 3 more coins each, Alex will still have less coins than Billy.

Example

Likewise

• If a < b, then a − c < b − c

• If a > b, then a + c > b + c, and

• If a > b, then a − c > b − c

So adding (or subtracting) the same value to both a and b will not change the inequality

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

3. If a < b, and c is negative, then ac > bc (inequality swaps over!)

This is true for division also !

• If a < b, and c is positive, then a < b c c

If a < b, and c is negative, then a > b

c c

• Who can provide an example of this inequality?

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

If a < b and b < c, then a < c

1.) If Alex is younger than Billy and

2.) Billy is younger than Carol,

Then Alex must be younger than Carol also!

Example

5. If a = b + c and b & c are > 0, then a > b and a > c

The Exterior Angle Inequality Theorem

• The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

1 43

2m 4 > m 1

m 4 > m 2

• Remember the exterior angle theorem? Based on the diagram, mL4 = _____ + ______

Remote time

If a and b are real numbers and a < b, which one of the following must be

true?

A. -a < -b

B. -a > -b

C. a < -b

D. -a > b

E. I don’t know

True or False

• If XY = YZ + 15, then XY > YZ

• If m A = m B + m C, then m B > m C

• If m H = m J+ m K, then m K > m H

• If 10 = y + 2, then y > 10

White Board Practice

Given: RS < ST; ST< RT

Conclusion: RS ___ RT

S T

R

White Board Practice

Given: RS < ST; ST< RT

Conclusion: RS < RT

S T

R

White Board Practice

Given: m PQU = m PQT + m TQU

Conclusion: m PQU ____ m TQU

m PQU ____ m PQT

U T

PR Q

White Board Practice

Given: m PQU = m PQT + m TQU

Conclusion: m PQU > m TQU

m PQU > m PQT

U T

PR 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

IF

THEN

ALL IF/THEN STATEMENTS CAN BE SHOWN USING A VENN DIAGRAM.

• 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

Venn Diagrams

Maria gets home from the game late

She will be grounded

If Maria gets home from the football game late, then she will be grounded.

Venn Diagrams

Mike eats three happy meals

He will have a major stomach ache

If Mike eats three happy meals, then he will have a major stomach ache.

Venn Diagrams

IF p,

THEN q.

If we have a true conditional statement, then we know that the hypothesis leads to the conclusion.

Summary of If-Then StatementsStatement 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

THESE STATEMENTS ARE EITHER BOTH TRUE OR BOTH FALSE!!!

• Inverse

• Converse

Mike eats three happy meals

He will have a major stomach ache

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.

Conditional / ContrapostiveLogically Equivalent

It’s 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

THEN

IF

What do the other colored circles represent?

Then she is grounded

Late from football game

Aren’t there other reasons why Maria might get grounded?

Has a major stomach ache

Eats three happy meals

Aren’t there other reasons why Mike might get a stomach ache?

Example 1

If it is snowing, then the game is canceled.

What can you conclude if I say, the game was cancelled?

Example 1

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

Snowing

AB

C

D

• All you can conclude it that it MIGHT be snowing and that isn’t much of a conclusion.

Let’s try again

• Remember don’t think logically. Think about where to put the star in the venn diagram.

Example 2

If you are in Coach Goss’s class, then you have homework every night.

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

Homework every night

Coach Goss’s class

AB

C

D

Jim might be in Coach Goss’s classNo Conclusion

Example 3

If you are in coach Goss’s class, then you have homework every night.

b) What can you conclude if I tell you Rob is in my 2nd period?

Homework every night

Coach Goss’s class

AB

C

D

Rob has homework every night

Example 4

If you are in Coach Goss’s class, then you have homework every night.

b) What can you conclude if I tell you Bill has Mr. Brady?

Homework every night

Coach Goss’s class

AB

C

D

Bill might have homework every nightNo conclusion

E

Example 5

If you are in coach Goss’s class, then you have homework every night.

d) What can you conclude if I tell you Matt never has homework?

Homework every night

Coach Goss’s class

AB

C

D

Matt is not in my class

E

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

We go on a picnic

Sun shines

a) We go on a picnic no conclusionb) The sun shines We go on a

picnicc) It is raining no

conclusiond) We do not go on a picnic

The sun is not shining

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

You are an athlete

Runner

a) Leroy is a runner He is an athleteb) Lucy is not an athlete She is not a runnerc) Linda is an athlete no conclusiond) Larry is not a runner no

conclusion

Warm - up

1. Write a statement that is logically equivalent to the following…

• “If it is Sunday, then John takes the trash out.”

2. 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

1. Write a statement that is logically equivalent to the following…

• “If it is Sunday, then John takes the trash out.”

• “If Bob can’t 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.

• 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?

a) I don’t like them at all

b) I don’t mind doing them

c) I haven’t learned all of the definitions/postulates/ and theorems, so they are still hard for me to do.

d) I love doing proofs

e) I’m 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 can’t 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 wasn’t 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 can’t 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.

Hudson’s 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 girlfriend’s 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 GIRLFRIEND’S 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.

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.

Given: Hudson drove 510 miles to the river in 6 hours.

Prove: 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 SR

Prove: PQ SR

Step 1: Assume temporarily PQ =SR

Given: Trapezoid PQRS with bases PQ and SR

Prove: PQ SR Step 1: Assume temporarily PQ =SRStep 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 can’t be a trapezoid AND a parallelogram.*

Given: Trapezoid PQRS with bases PQ and SR

Prove: PQ SR Step 1: Assume temporarily PQ =SRStep 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 can’t be a trapezoid AND a parallelogram.*

Step 3: My temporary assumption is false and PQ SR

Given: Trapezoid PQRS with bases PQ and SR

Prove: 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 can’t 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 anglesAssume 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 > x, then x < y. • If g < t and g > r , then t > r. • If c < r and c = f , then r > 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?

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. AC

B

AB > BC

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.

15 12this angle is larger than the other angle

White Board Practice

• Name the largest angle and the smallest angle of the triangle.

6

8

10

H

I J

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.

What do you think we can conclude if this angle measurement is greater than the other angle measurement?

A

B

C

White Board Practice

• Name the largest side and the shortest side of the triangle.

10546

T

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…….

VEGAS

LA

SF

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

c

b

a + b > c

a + c > b

b + c > a

**Always label your sides a, b, c and then just follow the rules of the theorem.**

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 > c

13 + c > 8

8 + c > 13

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 > c 13 + c > 8 8 + c > 13

21 > c c > -5 c > 5

c < 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 > c 13 + c > 8 8 + c > 13

21 > c c > -5 c > 5

c < 21

The 3rd side must be greater than the difference of the 2 #’sThe 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 ?

16 + 11 > 5 16 + 5 > 11 5 + 11 > 16

27 > 5 21 > 11 16 > 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

B E

C DFA

SSS Inequality Theorem

A

B E

C DF

White Board Practice

• Given: D is the midpoint of AC; m 1< m 2

What can you deduce?

A

B

12

DC

Complete with <, =, or >

m 1_ > _ m 2

4

3

21

4

4

Whiteboard Practice

• Page 230 – #1

• LCAB > LFDE [SSS ineq.]

– #3• RT > TS [SAS ineq.]

– #7• L1 > 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 ReviewSection 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