Transcript Chapter 1

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 6
Inequalities 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
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.
If a < b, then a + c < b + c
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
Example
1.) If Alex is younger than Billy and
2.) Billy is younger than Carol,
Then Alex must be younger than Carol also!
If a < b and b < c, then a < c
5. If a = b + c and b & c are > 0,
then a > b and a > c
The Exterior Angle Inequality
Theorem
• The
Remember
measure
theofexterior
an exterior
angle
angle
theorem?
of a Based
triangle
on
the diagram,
is greater
mthan
L4 = _____
the measure
+ ______
of either
remote interior angle.
m4>m1
2
m4>m2
1
3
4
Remote time
If a and b are real numbers and a < b,
which one of the following must be
true?
A.
B.
C.
D.
E.
-a < -b
-a > -b
a < -b
-a > b
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
R
S
T
White Board Practice
Given: RS < ST; ST< RT
Conclusion: RS < RT
R
S
T
White Board Practice
Given: m  PQU = m PQT + m TQU
Conclusion: m  PQU ____ m TQU
m  PQU ____ m PQT
U
R
T
Q
P
White Board Practice
Given: m  PQU = m PQT + m TQU
Conclusion: m  PQU > m TQU
m  PQU > m PQT
U
R
T
Q
P
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.
She will be grounded
Maria gets
home from
the game
late
Venn Diagrams
If Mike eats three happy meals, then he will have a major
stomach ache.
He will have a major
stomach ache
Mike eats
three
happy
meals
Venn Diagrams
If we have a true conditional statement, then we know that the
hypothesis leads to the conclusion.
THEN q.
IF p,
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 / Contrapostive
Logically Equivalent
He will have a major
stomach ache
Mike eats
three
happy
meals
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.
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
What do the other colored circles
represent?
THEN
IF
Aren’t there other reasons why Maria might get grounded?
Then she is grounded
Late from
football
game
Aren’t 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
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 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?
Jim might be in Coach Goss’s class
No Conclusion
Homework every night
B
D
CoachA
Goss’s
class
C
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?
Rob has homework every night
Homework every night
B
D
CoachA
Goss’s
class
C
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?
Bill might have homework every night
No conclusion
E
Homework every night
B
D
CoachA
Goss’s
class
C
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?
Matt is not in my class
E
Homework every night
B
D
CoachA
Goss’s
class
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
d) 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)
b)
c)
d)
Leroy is a runner
Lucy is not an athlete
Linda is an athlete
Larry is not a runner
You are an athlete
Runner
He is an athlete
She is not a runner
no conclusion
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.
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 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 =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 can’t be a
trapezoid AND a parallelogram.*
Given: Trapezoid PQRS with bases
PQ and SR
Prove: 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 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  Y
Prove:  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 = 80º
Prove:  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?
B
AB > BC
A
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.
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.
15
12
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
8
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.
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
46
R
105
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…….
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>c
a
a+c>b
b+c>a
c
b
**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
13 + 8 > c
13 + c > 8
c
8
8 + c > 13
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
21 > c
c > -5
c < 21
8 + c > 13
c>5
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
21 > c
c > -5
8 + c > 13
c>5
c < 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 ?
16 + 11 > 5
16 + 5 > 11
27 > 5
21 > 11
5 + 11 > 16
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
E
B
A
C
F
D
SSS Inequality Theorem
B
A
E
C
F
D
White Board Practice
• Given: D is the midpoint of AC; m  1< m
2
B
What can you deduce?
A
2 1
D
C
Complete with <, =, or >
m  1_ > _ m  2
4
4
1
2
4
3
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 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