Warm-up

1
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.
2
Chapter 6Inequalities in Geometry
3
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.

4
Law of Trichotomy

• The "Law of Trichotomy" says that only one of the
  following is true

5

• 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 !
6
Equalities vs Inequalities

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

7
Equalities vs Inequalities

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

8
The 4 Inequalities
Symbol Words
gt greater than
lt less than
greater than or equal to
less than or equal to
9
The symbol "points at" the smaller value
Does a lt b mean the same as b gt a?
10
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

11
1. If a lt b, then a c lt b c
12
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.

13
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
14
2. If a lt b, and c is positive, then ac lt bc
15
3. If a lt b, and c is negative, then ac gt bc
(inequality swaps over!)
16
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?
  
  

17
4. If a lt b and b lt c, then a lt c
18
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!

19
5. If a b c and b c are gt 0, then a gt b
and a gt c
Why ?
20
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
21
Remote time
22
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

23
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

24
White Board Practice

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

R
S
T
25
White Board Practice

• Given RS lt ST STlt RT
• Conclusion RS lt RT

R
S
T
26
White Board Practice

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

U
T
P
R
Q
27
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
28
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.

29
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.

30
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
31
Venn Diagrams
If Maria gets home from the football game late,
then she will be grounded.
32
Venn Diagrams
If Mike eats three happy meals, then he will have
a major stomach ache.
33
Venn Diagrams
If we have a true conditional statement, then we
know that the hypothesis leads to the conclusion.
34
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.
35
Logically Equivalent

• Conditional
• Contrapositve

• Inverse
• Converse

THESE STATEMENTS ARE EITHER BOTH TRUE OR BOTH
FALSE!!!
36
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.
37
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

38
Venn Diagrams
What do the other colored circles represent?
THEN
IF
39
Arent there other reasons why Maria might get
grounded?
Then she is grounded
Late from football game
40
Arent there other reasons why Mike might get a
stomach ache?
Has a major stomach ache
Eats three happy meals
41
Example 1

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

42
Example 1
Nothing !

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

43
There are other reasons that the game would be
cancelled
Game cancelled
D
B
Snowing
A
C
44

• All you can conclude it that it MIGHT be snowing
  and that isnt much of a conclusion.

45
Lets try again

• Remember dont think logically. Think about
  where to put the star in the venn diagram.

46
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?

47
Jim might be in Coach Gosss classNo Conclusion
Homework every night
D
A
Coach Gosss class
B
C
48
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?

49
Rob has homework every night
Homework every night
D
A
Coach Gosss class
B
C
50
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?

51
Bill might have homework every nightNo conclusion
E
Homework every night
D
A
Coach Gosss class
B
C
52
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?

53
Matt is not in my class
E
Homework every night
D
A
Coach Gosss class
B
C
54
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

55

• 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
56
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

57
First the statement MUST be in the form
if________, then_______

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

58
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

59
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
60
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

61
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

62
6-3 Indirect Proof

• Objectives
• Write indirect proofs in paragraph form

63

• After walking home, Sue enters the house carrying
  a dry umbrella.
• We can conclude that it is not raining outside.

Why?
64

• Because if it HAD been raining, then her umbrella
  would be wet.
• The umbrella is not wet.
• Therefore, it is not raining.

65
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

66
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.

67
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

68
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.

69
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.

70
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????

71
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.

72
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.

73
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.

74
Example 3

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

75
Given Trapezoid PQRS with bases PQ and
SRProve PQ ?SR

• Step 1 Assume temporarily PQ SR

76
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.

77
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

78
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

79
White board practice

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

80
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

81
White board practice

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

82
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.

83
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

84

• 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

85

• 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.

86
6-4 Inequalities for One Triangle

• Objectives
• State and apply the inequality theorems and
  corollaries for one triangle.

87
Remember the Isosceles Triangle Theorem

• If two sides of a triangle are congruent then the
  angles opposite those sides are congruent.

88
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
89
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
90
White Board Practice

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

H
10
6
I
J
8
91
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
92
White Board Practice

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

T
105
46
R
S
93
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.
94
Corollary 2

• The perpendicular segment from a point to a plane
  in the shortest segment from the point to the
  plane.

95

• Who knows the old saying The shortest distance
  between two points is.

A STRAIGHT LINE!!
96
SF
VEGAS
LA
97
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
98
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
99
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
100
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
101
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
102
White Board Practice

• Is it possible for a triangle to have sides with
  the lengths indicated?
• Yes No

103
6, 8, 10
Yes
104
3, 4, 8
No
105
4, 6, 2
No
106
6, 6, 5
Yes
107
Warm Up

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

108
6-5 Inequalities for Two Triangles

• Objectives
• State and apply the inequality theorems for two
  triangles

109
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

110
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.

111
Triangle Experiment

• Step 3 Now, INCREASE the size of the angle
  created by the two pencils. Draw another line
  connecting the open ends.

112
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

113
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.

114
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.

115
SAS Inequality Theorem
E
B
C
D
F
A
116
SSS Inequality Theorem
B
E
C
D
A
F
117
White Board Practice

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

B
1
2
A
C
D
118
Complete with lt, , or gt

• m ? 1_ gt _ m ? 2

4
4
1
2
3
4
119
Whiteboard Practice

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

120
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!

121
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