inequalities and indirect proofs in geometry honors...inequality involving the lengths of the sides...
TRANSCRIPT
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Inequalities
And
Indirect Proofs
In Geometry
Honors
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Table of Contents
DAY 1: (Ch 15-1) SWBAT: Write simple Inequality Proofs Pgs: 1-7 HW: 8-10 in packet
DAY 2: (Ch. 15-2) SWBAT: Write Inequality Proofs using Inequalities in Triangles Pgs: 11-15
HW: 16-18 in packet
DAY 3: (Ch. 15-1 to 15-2) SWBAT: Practice Writing Inequality Proofs Pgs: 19-22
HW: Finish the section
DAY 4: (5-1) SWBAT: Write Indirect Proofs
Pgs: 23-26
HW: 27-29 in packet
DAY 5: (5-1) SWBAT: More Practice Writing Indirect Proofs
Pgs: 30-32
HW: Finish the section
DAY 6: Review
Day 7: Test
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Inequality Postulates and Theorems
Postulate #1: A whole is greater than each of its parts.
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Transitive Property of Inequality
Postulate #2:
Substitution Postulate of Inequality
Postulate #3:
Model Problems
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Addition Postulate of Inequality
Postulate #4:
Postulate #5:
Model Problems
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Subtraction Postulate of Inequality:
Postulate #6:
Model Problems
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Multiplication Postulate of Inequality:
Postulate #7:
Division Postulate of Inequality:
Postulate #8:
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Model Problems
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SUMMARY
Method 1: Method 2:
(Addition Postulate of Inequality) (1, 2)
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Homework
6.
7.
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E is the midpoint of , F is the midpoint of
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Day 2 - Inequalities involving Triangles
Warm – Up
Inequality involving the lengths of the sides of a triangle
Postulate #9:
Inequalities involving the exterior angle of a triangle
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Postulate #10:
Postulate #11:
Postulate #12:
Model Problem
1.
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2.
Postulate #13
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Postulate #14
Model Problems
3.
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4.
SUMMARY
1. fdfd
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2.
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Homework
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Day 3 - Practice With inequality Proofs
Warm - Up
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1. Given:
2. Given: 1 2
Prove: RM > MS
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T
R SK M
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3.
4.
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5.
6.
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Day 4 - Indirect Proofs – Proof By Contradiction
When trying to prove a statement is true, it may be beneficial to ask yourself,
"What if this statement was not true?" and examine what happens. This is the
premise of the Indirect Proof or Proof by Contradiction.
Indirect Proof: Assume what you need to prove is false, and then show that something
contradictory (absurd) happens.
Steps in an Indirect Proof:
Assume that the opposite of what you are trying to prove is true.
From this assumption, see what conclusions can be drawn. These
conclusions must be based upon the assumption and the use of valid
statements.
Search for a conclusion that you know is false because it contradicts
given or known information. Oftentimes you will be contradicting a
piece of GIVEN information.
Since your assumption leads to a false conclusion, the assumption must
be false.
If the assumption (which is the opposite of what you are trying to prove)
is false, then you will know that what you are trying to prove must be
true.
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Example #1
Given: m A = 50 and mB = 70
Prove: A and B are not complementary
Example #2
Given :
Prove: and do not bisect each other.
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3. Given: ΔABC is scalene
BD bisects ABC.
Prove: BD is not perpendicular to AC
4.
A
B
C D
1 2
3 4
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SUMMARY
Assumption leading to a Contradiction
9. B is not to CED 9. Contradiction (7, 8)
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Homework
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4. Given: is scalene
Prove:
5. Given: BE is the median of AC ,
Prove: ΔABC is not Isosceles
CBEABE
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6.
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Day 5 - More Indirect Proofs
Prove each of the following indirectly.
1. Given:
BAC DAC
Prove:
2. Given: l m
Prove: 1 2
A
BDC
m
l
t
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3.
4. Prove that if ABC is isosceles with base and if P is a point on that is not
the midpoint, and then does not bisect BAC.
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5. Given: 1 2
ABCD is not a parallelogram
Prove: 3 4
6. Given: O
is not an altitude
Prove: does not bisect AOC
O
CA
B