The document provides information on algebraic proofs and two-column proofs. It defines an algebraic proof as using a series of algebraic steps to solve problems and justify steps. A two-column proof is defined as having one column for statements and a second column for justifying each statement. Properties of equality like addition, subtraction, multiplication, and division properties are also defined for writing algebraic proofs. An example problem is worked through step-by-step to demonstrate an algebraic proof.

1. Section 2-6
Algebraic Proof
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2. Essential Questions
How do you use Algebra to write two-column proofs?
How do you use properties of equality to write geometric
proofs?
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3. Vocabulary
1. Algebraic Proof:
2. Two-column Proof:
3. Formal Proof:
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4. Vocabulary
1. Algebraic Proof: When a series of algebraic steps are used to solve
problems and justify steps.
2. Two-column Proof:
3. Formal Proof:
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5. Vocabulary
1. Algebraic Proof: When a series of algebraic steps are used to solve
problems and justify steps.
2. Two-column Proof: A format for a proof where one column provides
statements of what is being done, and the second is to justify each
statement
3. Formal Proof:
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6. Vocabulary
1. Algebraic Proof: When a series of algebraic steps are used to solve
problems and justify steps.
2. Two-column Proof: A format for a proof where one column provides
statements of what is being done, and the second is to justify each
statement
3. Formal Proof: Another name for a two-column proof
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7. Properties of Real Numbers
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8. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

9. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

10. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

11. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

12. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

13. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

14. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

15. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

16. Properties of Real Numbers
Addition Property of Equality:
If a = b, then a + c = b + c
Subtraction Property of Equality:
If a = b, then a − c = b − c
Multiplication Property of Equality:
If a = b, then a × c = b × c
Division Property of Equality:
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property of Equality:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Transitive Property of Equality:
If a = b and b = c, then a = c
If a = b, then a may be replaced by b
Substitution Property of Equality:
in any equation/expression
Distributive Property:
Wednesday, November 6, 13
a(b + c) = ab + ac

17. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
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18. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
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Given

19. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
10 − 6a − 4a − 28 = 92
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Given

20. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
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21. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
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22. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
Substitution Property
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23. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
+18 +18
Substitution Property
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24. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
+18 +18
Substitution Property
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Addition Property

25. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
+18 +18
Substitution Property
−10 a = 110
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Addition Property

26. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
+18 +18
Substitution Property
−10 a = 110
Substitution Property
Wednesday, November 6, 13
Addition Property

27. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
+18 +18
Substitution Property
−10 a = 110
−10 −10
Substitution Property
Wednesday, November 6, 13
Addition Property

28. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
+18 +18
Substitution Property
−10 a = 110
−10 −10
Substitution Property
Division Property
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Addition Property

29. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
+18 +18
Substitution Property
−10 a = 110
−10 −10
Substitution Property
Division Property
a = −11
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Addition Property

30. Example 1
Solve the equation. Write a justification for each step.
2(5−3a)− 4(a +7) = 92
Given
10 − 6a − 4a − 28 = 92
Distributive Property
−10a −18 = 92
+18 +18
Substitution Property
−10 a = 110
−10 −10
Substitution Property
Division Property
a = −11
Substitution Property
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Addition Property

31. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
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32. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
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Reasons

33. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1.
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Reasons

34. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
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Reasons

35. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
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Reasons
Given

36. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2.
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Reasons
Given

37. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2. d −5 = 20t
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Reasons
Given

38. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2. d −5 = 20t
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Reasons
Given
Subtraction Property

39. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2. d −5 = 20t
3.
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Reasons
Given
Subtraction Property

40. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2. d −5 = 20t
d−5
=t
3.
20
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Reasons
Given
Subtraction Property

41. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2. d −5 = 20t
d−5
=t
3.
20
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Reasons
Given
Subtraction Property
Division Property

42. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2. d −5 = 20t
d−5
=t
3.
20
4.
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Reasons
Given
Subtraction Property
Division Property

43. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2. d −5 = 20t
d−5
=t
3.
20
4.
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d −5
t=
20
Reasons
Given
Subtraction Property
Division Property

44. Example 2
If the distance d an object travels is given by d = 20t +5,
d −5
the time t that the object travels is given by t =
.
20
Write a two-column proof to verify this conjecture.
Statements
1. d = 20t +5
2. d −5 = 20t
d−5
=t
3.
20
4.
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d −5
t=
20
Reasons
Given
Subtraction Property
Division Property
Symmetric Property

45. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Wednesday, November 6, 13

46. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
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Reasons

47. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1.
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Reasons

48. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
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Reasons

49. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
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Reasons
Given

50. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
2.
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Reasons
Given

51. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
2.
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m∠A = m∠B
Reasons
Given

52. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
2.
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m∠A = m∠B
Reasons
Given
Def. of ≅ angles

53. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
2.
3.
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m∠A = m∠B
Reasons
Given
Def. of ≅ angles

54. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
2.
m∠A = m∠B
3.
m∠A = 2m∠C
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Reasons
Given
Def. of ≅ angles

55. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
Reasons
Given
2.
m∠A = m∠B
Def. of ≅ angles
3.
m∠A = 2m∠C
Transitive Property
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56. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
Reasons
Given
2.
m∠A = m∠B
Def. of ≅ angles
3.
m∠A = 2m∠C
Transitive Property
4.
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57. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
Reasons
Given
2.
m∠A = m∠B
Def. of ≅ angles
3.
m∠A = 2m∠C
Transitive Property
4.
m∠A = 2(45°)
Wednesday, November 6, 13

58. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
Reasons
Given
2.
m∠A = m∠B
Def. of ≅ angles
3.
m∠A = 2m∠C
Transitive Property
4.
m∠A = 2(45°)
Substitution Property
Wednesday, November 6, 13

59. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
Reasons
Given
2.
m∠A = m∠B
Def. of ≅ angles
3.
m∠A = 2m∠C
Transitive Property
4.
m∠A = 2(45°)
Substitution Property
5.
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60. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
Reasons
Given
2.
m∠A = m∠B
Def. of ≅ angles
3.
m∠A = 2m∠C
Transitive Property
4.
m∠A = 2(45°)
Substitution Property
5.
m∠A = 90°
Wednesday, November 6, 13

61. Example 3
If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°.
Write a two-column proof to verify this conjecture.
Statements
1. ∠A ≅ ∠B, m∠B = 2m∠C,
and m∠C = 45°
Reasons
Given
2.
m∠A = m∠B
Def. of ≅ angles
3.
m∠A = 2m∠C
Transitive Property
4.
m∠A = 2(45°)
Substitution Property
5.
m∠A = 90°
Substitution Property
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62. Problem Set
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63. Problem Set
p. 137 #1-27 odd, 36
“The greatest challenge to any thinker is stating the problem in a way that
will allow a solution.” - Bertrand Russell
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