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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

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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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Homework

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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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Homework

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19. Given: C is the midpoint of BD 1 2m m∠ = ∠ m ∠3 > m∠4

Prove: AB > ED

20. Given: ∠SRT ≅ ∠STR TU > RU

Prove: m TSU m RSU∠ > ∠

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Practice With inequality Proofs

1. Given:

2. Given: 𝑅𝑇���� ≅ 𝑆𝑇���� ∠1 ≅ ∠2 Prove: RM > MS

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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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3. Given: ΔABC is scalene BD bisects ∠ABC. Prove: BD is not perpendicular to AC

A

B

C D

1 2

3 4

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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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More Indirect Proofs

Prove each of the following indirectly. 1. Given: 𝐴𝐵���� ≅ 𝐴𝐷���� ∠BAC ≅ ∠DAC Prove: 𝐵𝐶���� ≅ 𝐷𝐶���� 2. Given: l // m

Prove: ∠1 ≅ ∠2

m

l

t

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3. Given: ∠ D ≅ ∠ABE 𝐵𝐸���� // 𝐶𝐷���� Prove: 𝐴𝐶���� ≅ 𝐴𝐷���� 4. Prove that if ∆ABC is isosceles with base 𝐵𝐶���� and if P is a point on 𝐵𝐶 that is not

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 𝑂𝐵���� is not an altitude Prove: 𝑂𝐵���� does not bisect ∡AOC

O

CA B