The document discusses writing proofs in geometry. It begins by defining key terms used in proofs such as postulate, axiom, proof, and flow proof. It then provides examples of paragraph and algebraic proofs and discusses using properties of equality and substitution in proofs. The document emphasizes analyzing geometric figures using postulates and writing logical arguments to construct proofs.

1. SECTION 2-4
Writing Proofs

2. ESSENTIAL QUESTIONS
➤How do you analyze figures to identify and use
postulates about points, lines, and planes?
➤How do you analyze and construct viable arguments in
proofs?

3. VOCABULARY
1. Postulate:
2. Axiom:
3. Proof:
4. Flow Proof:
5. Deductive Argument:

4. VOCABULARY
1. Postulate: A statement that is accepted to be true
without proof
2. Axiom:
3. Proof:
4. Flow Proof:
5. Deductive Argument:

5. VOCABULARY
1. Postulate: A statement that is accepted to be true
without proof
2. Axiom: Another name for a postulate
3. Proof:
4. Flow Proof:
5. Deductive Argument:

6. VOCABULARY
1. Postulate: A statement that is accepted to be true
without proof
2. Axiom: Another name for a postulate
3. Proof: A logical argument made up of statements that
are supported by another statement that is accepted as
true
4. Flow Proof:
5. Deductive Argument:

7. VOCABULARY
1. Postulate: A statement that is accepted to be true
without proof
2. Axiom: Another name for a postulate
3. Proof: A logical argument made up of statements that
are supported by another statement that is accepted as
true
4. Flow Proof: Uses statements in boxes with arrows to
show the logical progression of an argument
5. Deductive Argument:

8. VOCABULARY
1. Postulate: A statement that is accepted to be true
without proof
2. Axiom: Another name for a postulate
3. Proof: A logical argument made up of statements that
are supported by another statement that is accepted as
true
4. Flow Proof:
A logical chain of statements
that link the given to what you are trying to prove
Uses statements in boxes with arrows to
show the logical progression of an argument
5. Deductive Argument:

9. VOCABULARY
6. Algebraic Proof:
7. Theorem:
8. Paragraph Proof:
9. Informal Proof:

10. VOCABULARY
6. Algebraic Proof: When a series of algebraic steps are
used to solve problems and justify steps.
7. Theorem:
8. Paragraph Proof:
9. Informal Proof:

11. VOCABULARY
6. Algebraic Proof: When a series of algebraic steps are
used to solve problems and justify steps.
7. Theorem:
8. Paragraph Proof:
9. Informal Proof:
A statement or conjecture that has been
proven true

12. VOCABULARY
6. Algebraic Proof: When a series of algebraic steps are
used to solve problems and justify steps.
7. Theorem:
8. Paragraph Proof: When a paragraph is written to
logically explain why a given conjecture is true
9. Informal Proof:
A statement or conjecture that has been
proven true

13. VOCABULARY
6. Algebraic Proof: When a series of algebraic steps are
used to solve problems and justify steps.
7. Theorem:
8. Paragraph Proof: When a paragraph is written to
logically explain why a given conjecture is true
9. Informal Proof:
A statement or conjecture that has been
proven true
Another name for a paragraph proof
as it allows for free writing to provide the logical
explanation

14. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!

15. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!
2.1: Through any two points, there is exactly one line.

16. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!
2.1: Through any two points, there is exactly one line.
2.2: Through any three noncollinear points, there is exactly one
plane.

17. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!
2.1: Through any two points, there is exactly one line.
2.2: Through any three noncollinear points, there is exactly one
plane.
2.3: A line contains at least two points.

18. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!
2.1: Through any two points, there is exactly one line.
2.2: Through any three noncollinear points, there is exactly one
plane.
2.3: A line contains at least two points.
2.4: A plane contains at least three noncollinear points.

19. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!
2.1: Through any two points, there is exactly one line.
2.2: Through any three noncollinear points, there is exactly one
plane.
2.3: A line contains at least two points.
2.4: A plane contains at least three noncollinear points.
2.5: If two points lie in a plane, then the entire line containing
those points lies in the plane.

20. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!

21. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!
2.6: If two lines intersect, then their intersection is exactly one
point.

22. HARKENING BACK TO CHAPTER 1
Old ideas about points, lines, and planes are now postulates!
2.6: If two lines intersect, then their intersection is exactly one
point.
2.7: If two planes intersect, then their intersection is a line.

23. EXAMPLE 1
Determine whether the statement is always, sometimes, or
never true.
a. Points E and F are contained by exactly one line.
b. There is exactly one plane that contains points A, B, and C.

24. EXAMPLE 1
Determine whether the statement is always, sometimes, or
never true.
a. Points E and F are contained by exactly one line.
Always true
b. There is exactly one plane that contains points A, B, and C.

25. EXAMPLE 1
Determine whether the statement is always, sometimes, or
never true.
a. Points E and F are contained by exactly one line.
Always true
Only one line can be drawn through any two points
b. There is exactly one plane that contains points A, B, and C.

26. EXAMPLE 1
Determine whether the statement is always, sometimes, or
never true.
a. Points E and F are contained by exactly one line.
Always true
Only one line can be drawn through any two points
b. There is exactly one plane that contains points A, B, and C.
Sometimes true

27. EXAMPLE 1
Determine whether the statement is always, sometimes, or
never true.
a. Points E and F are contained by exactly one line.
Always true
Only one line can be drawn through any two points
b. There is exactly one plane that contains points A, B, and C.
Sometimes true
If the three points are collinear, then an infinite number
planes can be drawn. If they are noncollinear, then it is true.

28. EXAMPLE 1
Determine whether the statement is always, sometimes, or
never true.
c. Planes R and T intersect at point P.

29. EXAMPLE 1
Determine whether the statement is always, sometimes, or
never true.
c. Planes R and T intersect at point P.
Never true

30. EXAMPLE 1
Determine whether the statement is always, sometimes, or
never true.
c. Planes R and T intersect at point P.
Never true
Two planes intersect in a line

31. EXAMPLE 2
Given that AC intersects CD, write a paragraph proof to show
that A, C, and D determine a plane.

32. EXAMPLE 2
Given that AC intersects CD, write a paragraph proof to show
that A, C, and D determine a plane.
The two lines intersect, so they must intersect at point
C because two lines intersect in exactly one point.

33. EXAMPLE 2
Given that AC intersects CD, write a paragraph proof to show
that A, C, and D determine a plane.
The two lines intersect, so they must intersect at point
C because two lines intersect in exactly one point.
Points A and D are on different lines, so A, C, and D
are noncollinear by definition of noncollinear.

34. EXAMPLE 2
Given that AC intersects CD, write a paragraph proof to show
that A, C, and D determine a plane.
The two lines intersect, so they must intersect at point
C because two lines intersect in exactly one point.
Points A and D are on different lines, so A, C, and D
are noncollinear by definition of noncollinear.
Points A, C, and D determine a plane because three
noncollinear points determine exactly one plane.

35. EXAMPLE 3
Given that M is the midpoint of XY, write a paragraph proof to
show that XM ≅ MY.

36. EXAMPLE 3
If M is the midpoint of XY, then by the definition of midpoint,
XM = MY. Since they have the same measure, we know that,
by the definition of congruence, XM ≅ MY.
Given that M is the midpoint of XY, write a paragraph proof to
show that XM ≅ MY.

37. EXAMPLE 3
If M is the midpoint of XY, then by the definition of midpoint,
XM = MY. Since they have the same measure, we know that,
by the definition of congruence, XM ≅ MY.
Theorem 2.1 (Midpoint Theorem):
Given that M is the midpoint of XY, write a paragraph proof to
show that XM ≅ MY.

38. EXAMPLE 3
If M is the midpoint of XY, then by the definition of midpoint,
XM = MY. Since they have the same measure, we know that,
by the definition of congruence, XM ≅ MY.
Theorem 2.1 (Midpoint Theorem): If M is the midpoint of
XY, then XM ≅ MY.
Given that M is the midpoint of XY, write a paragraph proof to
show that XM ≅ MY.

39. PROPERTIES OF REAL NUMBERS

40. PROPERTIES OF REAL NUMBERS

41. PROPERTIES OF REAL NUMBERS
Addition Property of Equality:

42. PROPERTIES OF REAL NUMBERS
Addition Property of Equality: If a = b, then a + c = b + c

43. PROPERTIES OF REAL NUMBERS
Addition Property of Equality: If a = b, then a + c = b + c
Subtraction Property of Equality:

44. 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

45. 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:

46. 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

47. 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:

48. 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

49. 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:

50. 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

51. 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:

52. 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

53. 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:

54. 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

55. 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
Substitution Property of Equality:

56. 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
Substitution Property of Equality:
If a = b, then a may be replaced by b in
any equation/expression

57. 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
Substitution Property of Equality:
If a = b, then a may be replaced by b in
any equation/expression
Distributive Property:

58. 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
Substitution Property of Equality:
If a = b, then a may be replaced by b in
any equation/expression
Distributive Property: a(b + c) = ab + ac

59. EXAMPLE 4
2(5− 3a)− 4(a + 7)= 92
Solve. Write a justification for each step.

60. EXAMPLE 4
2(5− 3a)− 4(a + 7)= 92
Solve. Write a justification for each step.
Given

61. EXAMPLE 4
2(5− 3a)− 4(a + 7)= 92
Solve. Write a justification for each step.
10 − 6a − 4a − 28 = 92
Given

62. EXAMPLE 4
2(5− 3a)− 4(a + 7)= 92
Solve. Write a justification for each step.
10 − 6a − 4a − 28 = 92
Given
Distributive Property

63. EXAMPLE 4
2(5− 3a)− 4(a + 7)= 92
Solve. Write a justification for each step.
10 − 6a − 4a − 28 = 92
−10a −18 = 92
Given
Distributive Property

64. EXAMPLE 4
2(5− 3a)− 4(a + 7)= 92
Solve. Write a justification for each step.
10 − 6a − 4a − 28 = 92
−10a −18 = 92
Given
Distributive Property
Substitution Property

65. EXAMPLE 4
2(5− 3a)− 4(a + 7)= 92
Solve. Write a justification for each step.
10 − 6a − 4a − 28 = 92
−10a −18 = 92
+18 +18
Given
Distributive Property
Substitution Property

66. EXAMPLE 4
2(5− 3a)− 4(a + 7)= 92
Solve. Write a justification for each step.
10 − 6a − 4a − 28 = 92
−10a −18 = 92
+18 +18
Given
Distributive Property
Substitution Property
Addition Property

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

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

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

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

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

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

73. EXAMPLE 5
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20

74. EXAMPLE 5
d = 20t + 5
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20

75. EXAMPLE 5
d = 20t + 5 Given
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20

76. EXAMPLE 5
d = 20t + 5
d − 5+ 20t
Given
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20

77. EXAMPLE 5
d = 20t + 5
d − 5+ 20t
Given
Subtraction Property
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20

78. EXAMPLE 5
d = 20t + 5
d − 5+ 20t
d − 5
20
= t
Given
Subtraction Property
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20

79. EXAMPLE 5
d = 20t + 5
d − 5+ 20t
d − 5
20
= t
Given
Subtraction Property
Division Property
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20

80. EXAMPLE 5
d = 20t + 5
d − 5+ 20t
d − 5
20
= t
Given
Subtraction Property
Division Property
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20
t =
d − 5
20

81. EXAMPLE 5
d = 20t + 5
d − 5+ 20t
d − 5
20
= t
Given
Subtraction Property
Division Property
If the distance d an object travels can be given by
, the time t that the object travels is given by
. . Write an algebraic proof to verify this conjecture.
d = 20t + 5
t =
d − 5
20
t =
d − 5
20
Symmetric Property