Name _________________________________ Per. ________ Ms. Williams – Geometry Honors
Inequalities
And
Indirect Proofs
In Geometry
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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
4
Subtraction Postulate of Inequality:
Postulate #6:
Model Problems
5
Multiplication Postulate of Inequality:
Postulate #7:
Division Postulate of Inequality:
Postulate #8:
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Model Problems
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Homework
6.
7.
8
9
10
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Inequalities involving Triangles
Inequality involving the lengths of the sides of a triangle
Postulate #9:
Inequalities involving the exterior angle of a triangle
Postulate #10:
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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.
Homework
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19. Given: C is the midpoint of BD
m∠1 = m∠2
m ∠3 > m∠4
Prove: AB > ED
20.
Given: ∠SRT ≅ ∠STR
TU > RU
Prove: m∠TSU > m∠RSU
21.
22.
22.
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Practice With inequality Proofs
1. Given:
2. Given: ����
𝑅𝑇 ≅ ����
𝑆𝑇 ∠1 ≅ ∠2
Prove: RM > MS
1
2
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3.
4.
19
5.
6.
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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 m∠B = 70°
Prove: ∠A and ∠B are not complementary
Example #2
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B
3. Given: ΔABC is scalene
BD bisects ∠ABC.
Prove: BD is not perpendicular to AC
1 2
A
3 4
D
C
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Homework
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4. Given:
is scalene
Prove:
5. Given: BE is the median of AC , ∠ABE
Prove: ΔABC is not Isosceles
≠ ∠CBE
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6.
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More Indirect Proofs
Prove each of the following indirectly.
1. Given: ����
𝐴𝐵 ≅ ����
𝐴𝐷
∠BAC ≅ ∠DAC
���� ≅ 𝐷𝐶
����
Prove: 𝐵𝐶
t
2. Given: l // m
Prove: ∠1 ≅ ∠2
1
l
2
m
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3. Given: ∠ D ≅ ∠ABE
���� // ����
𝐵𝐸
𝐶𝐷
����
����
Prove: 𝐴𝐶 ≅ 𝐴𝐷
���� and if P is a point on 𝐵𝐶 that is not
4. Prove that if ∆ABC is isosceles with base 𝐵𝐶
�����
the midpoint, and then 𝐴𝑃 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
𝑂𝐵
���� does not bisect ∡AOC
Prove: 𝑂𝐵
O
A
B
C
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