chapter 6 inequalities in geometry
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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
A review of some properties of inequalities
• 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
Likewise
• If a < b, and c is positive, then a < b c c
• So multiplying (or dividing) the same value to both a and b will no change the inequality if c is POSITIVE !
3. If a < b, and c is negative, then ac > bc (inequality swaps over!)
Likewise
• If a < b, and c is negative, then a > b
c c
• So multiplying (or dividing) the same value to both a and b will change the inequality if c is NEGATIVE !
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 c is > 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
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
Remote Time
• True or False
True or False
• If XY = YZ + 15, then XY > YZ
True or False
• If m A = m B + m C, then m B > m C
True or False
• If m H = m J+ m K, then m K > m H
True or False
• 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 converse and inverse of an if-then statement.
• Understand the relationship between logically equivalent statements.
• Draw correct conclusions from given statements.
Review
• Identify the hypothesis and the conclusion of each statements.– If Maria gets home from the football game late,
then she will be grounded.– If Mike eats three happy meals, then he will
have a major stomach ache.
• If you are in your room, then you are in your house.
What can you conclude if
a) You are in your house
b) You are in your room
c) You are not in your room
d) You are not in your house
Venn Diagrams
Maria gets home from the game late
She will be grounded
A conditional statement can also be illustrated with a Venn Diagram.
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
A conditional statement can also be illustrated with a Venn Diagram.
If Mike eats three happy meals, then he will have a major stomach ache
Venn Diagrams
IF
THEN
Venn Diagrams
THEN
IF
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?
Summary of If-Then Statements
Statement Formed by Symbols
Conditional Given hypothesis and conclusion
If p, then q
Converse Switching the hypothesis and the conclusion
If q, then p
Inverse Negating the hypothesis and the conclusion
If not p, then not q
Contrapositive Negating and switching the hypothesis and the conclusion
If not q, then not p
Logically Equivalent
Statement Formed by Symbols
Conditional Given hypothesis and conclusion
If p, then q
Contrapositive Negating and switching the hypothesis and the conclusion
If not q, then not p
These statements are either both true or both false
Summary of If-Then Statements
Statement Formed by Symbols
Converse Switching the hypothesis and the conclusion
If q, then p
Inverse Negating the hypothesis and the conclusion
If not p, then not q
These statements are either both true or both false
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
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 Ms. Vasquez class, then you have homework every night.
a) What can you conclude if I tell you Jim has homework every night?
Homework every night
Ms Vasquez class
AB
C
D
Jim might be in Ms. Vasquez classNo Conclusion
Example 2
If you are in Ms. Vasquez class, then you have homework every night.
b) What can you conclude if I tell you Rob is in my 6th period?
Homework every night
Ms Vasquez class
AB
C
D
Rob has homework every night
Example 2
If you are in Ms. Vasquez class, then you have homework every night.
b) What can you conclude if I tell you Bill has Mr. Brady
Homework every night
Ms Vasquez class
AB
C
D
Bill might have homework every nightNo conclusion
E
Example 2
If you are in Ms. Vasquez class, then you have homework every night.
d) What can you conclude if I tell you Matt never has homework?
Homework every night
Ms Vasquez class
AB
C
D
Matt is not in my class
E
White Board Practice
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
White Board Practice
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
AB
C
D
E
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
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
You are an athlete
Runner
AB
C
D
E
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
If a car has anti-lock brakes, then it must be relatively new.
• What can you conclude if
(a) This car is relatively new. (b) This car does not have anti-lock brakes. (c) This car is not new.
If it rains tomorrow, I'll pick you up for school.
• What can you conclude if
(a) It rains tomorrow. (b) I don't pick you up for school. (c) It does not rain tomorrow. (d) I pick you up for school.
If you own a Saturn, then you own a car.
• What can you conclude if
(a) You do not own a car. (b) You own a Honda. (c) You own a car.
What is the inverse of "If it is Saturday, then it is the
weekend"?A) If it is the weekend, then it is Saturday
B) If it is not Saturday, then it is the weekend
C) If it is not Saturday, then it is not the weekend
D) If it is not the weekend, then it is not Saturday
If you are a doctor, then you are a college graduate.
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
Step 1
• Assume temporarily that…. (the conclusion is false). I know I always tell you not to ASSume, but here you can. You want to believe that the opposite of the conclusion is true.
Step 2
• Using the given information of anything else that you already know for sure, for sure, 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.
Step 3
• My temporary assumption is false and ( the original conclusion must be true). Restate the original conclusion.
Example 1
Given: Tim drove 105 miles to his friend’s house in 1 ½ hours.
Prove: Tim exceeded the 55 mph speed limit while driving.
Given: Tim drove 105 miles to his friend’s house in 1 ½ hours.
Prove: Tim exceeded the 55 mph speed limit while driving.
Step 1: Assume temporarily that Tim did not exceed the 55 mph
Given: Tim drove 105 miles to his friend’s house in 1 ½ hours.
Prove: Tim exceeded the 55 mph speed limit while driving.
Step 1: Assume temporarily that Tim didnot exceed the 55 mphStep 2: Then the minimum time it would take
Tim to get to his friend’s house is 105/55 = 1.9 hours. This is a contradiction to the given information that he got there in 1 ½ hours.*
Given: Tim drove 105 miles to his friend’s house in 1 ½ hours.
Prove: Tim exceeded the 55 mph speed limit while driving.
Step 1: Assume temporarily that Tim didnot exceed the 55 mphStep 2: Then the minimum time it would take Tim to
get to his friend’s house is 105/55 = 1.9 hours. This is a contradiction to the given information that he got there in 1 ½ hours.*
Step 3: My temporary assumption is false and Tim exceeded the 55 mph speed limit while driving.
Given: Tim drove 105 miles to his friend’s house in 1 ½ hours.
Prove: Tim exceeded the 55 mph speed limit while driving.
Assume temporarily that Tim did not exceed the 55 mph. Then the minimum time it would take Tim to get to his friend’s house is 105/55 = 1.9 hours. This is a contradiction to the given information that he got there in 1 ½ hours.* My temporary assumption is false and Tim exceeded the 55 mph speed limit while driving.
Example 2
Given: n is an integer and n2 is even
Prove: n is even
Given: n is an integer and n2 is evenProve: n is even
Step 1: Assume temporarily that n is not even. That would mean that n is odd.
Given: n is an integer and n2 is evenProve: n is even
Step 1: Assume temporarily that n is not even. That would mean that n is odd.
Step 2: I know that n2 = (n)(n), and if I choose a value for n that is odd, like 3, then n2 =(3)(3)=9.* This contradicts the given information that is n2 even.
Given: n is an integer and n2 is evenProve: n is even
Step 1: Assume temporarily that n is not even. That would mean that n is odd.
Step 2: I know that n2 = (n)(n), and if I choose a value for n that is odd, like 3, then n2 =(3)(3)=9.* This contradicts the given information that is n2 even.
Step 3: My temporary assumption is false and n is even.
Given: n is an integer and n2 is evenProve: n is even
Assume temporarily that n is not even. That would mean that n is odd. I know that n2 = (n)(n), and if I choose a value for n that is odd, like 3, then n2 =(3)(3)=9.* This contradicts the given information that is n2 even. My temporary assumption is false and n is even.
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.
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?
Theorem 6-2
If one side of a triangle is longer than a second side, then the angle opposite the first side is longer than the angle opposite the second side.
White Board Practice
• Name the largest angle and the smallest angle of the triangle.
6
8
10
H
I J
Theorem 6-3
• 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.
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.
Corollary 2
• The perpendicular segment from a point to a plane in the shortest segment from the point to the plane.
Theorem 6-4The 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
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
White Board Practice
• Is it possible for a triangle to have sides with lengths 16, 11, 5 ?
16 + 11 > 5 16 + 5 > 11 5 + 1 > 16
27 > 5 21 > 11 6 > 16
Remote Time
• Is it possible for a triangle to have sides with the lengths indicated?
Yes No
6, 8, 10
3, 4, 8
2.5, 4.1, 5.0
4, 6, 2
6, 6, 5
6-5 Inequalities for Two Triangles
Objectives
• State and apply the inequality theorems for two triangles
Remember SAS and SSS
Paper Strip Triangle
Supplies- paper, scissors and a ruler.Step 1: Have pairs of students cut two
strips of paper, making the strips a random length but very thin.
Step 2: Students then place the two strips together so that they form two sides of an angle
Step 3: Then they measure how long the third side would need to be.
Paper Strip Triangle
Step 4: Now, have the students increase the size of the included angle and measure how long the third side would need to be.
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
Theorem 6-5SAS Inequality Theorem
• If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.
Theorem 6-5SSS Inequality Theorem
• If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second triangle.
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
Complete with <, =, or >
m 1____ m 2
4
3
21
4
4
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