1. Proving Statements in Geometry Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

2. Inductive Reasoning ERHS Math Geometry Mr. Chin-Sung Lin

3. Visual Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe and sketch the fourth figure in the pattern: Fig. 1 ? Fig. 2Fig. 3Fig. 4

4. Visual Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe and sketch the fourth figure in the pattern: Fig. 1Fig. 2Fig. 3Fig. 4

5. Visual Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe and sketch the fourth figure in the pattern: Fig. 1 ? Fig. 2Fig. 3Fig. 4

6. Visual Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe and sketch the fourth figure in the pattern: Fig. 1Fig. 2Fig. 3Fig. 4

7. Visual Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe and sketch the fourth figure in the pattern: Fig. 1 ? Fig. 2Fig. 3Fig. 4

8. Visual Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe and sketch the fourth figure in the pattern: Fig. 1Fig. 2Fig. 3Fig. 4

9. Visual Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe and sketch the fourth figure in the pattern: Fig. 1 ? Fig. 2Fig. 3Fig. 4

10. Visual Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe and sketch the fourth figure in the pattern: Fig. 1Fig. 2Fig. 3Fig. 4

11. Number Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe the pattern in the numbers and write the next three numbers: 1 ? 4 7 10 ? ?

12. Number Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe the pattern in the numbers and write the next three numbers: 1 4 7 10 13 16 19 3 3 3 3 3 3

13. Number Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe the pattern in the numbers and write the next three numbers: 1 ? 4 9 16 ? ?

14. Number Pattern ERHS Math Geometry Mr. Chin-Sung Lin Describe the pattern in the numbers and write the next three numbers: 25 36 49 3 5 7 9 11 13 1 4 9 16 2 2 2 2 2

15. Conjecture ERHS Math Geometry Mr. Chin-Sung Lin An unproven statement that is based on observation

16. Inductive Reasoning ERHS Math Geometry Mr. Chin-Sung Lin Inductive reasoning, or induction, is reasoning from a specific case or cases and deriving a general rule You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case

17. Weakness of Inductive Reasoning ERHS Math Geometry Mr. Chin-Sung Lin Direct measurement results can be only approximate We arrive at a generalization before we have examined every possible example When we conduct an experiment we do not give explanations for why things are true

18. Strength of Inductive Reasoning ERHS Math Geometry Mr. Chin-Sung Lin A powerful tool in discovering new mathematical facts (making conjectures) Inductive reasoning does not prove or explain conjectures

19. Make a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin Give five colinear points, make a conjecture about the number of ways to connect different pairs of the points

20. Make a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin Give five colinear points, make a conjecture about the number of ways to connect different pairs of the points No of Points 12345 Picture No of Connections 0136?

21. Make a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin No of Points 12345 Picture No of Connections 0136? 1 2 3 ?

22. Make a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin No of Points 12345 Picture No of Connections 013610 Conjecture: You can connect five colinear points 6 + 4 = 10 different ways 1 2 3 4

23. Prove a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin No of Points 12345 Picture No of Connections 013610 Conjecture: You can connect five colinear points 6 + 4 = 10 different ways 1 2 3 4

24. Prove a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin To show that a conjecture is true, you must show that it is true for all cases

25. Disprove a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin To show that a conjecture is false, you just need to find one counterexample A counterexample is a specific case for which the conjecture is false

26. Exercise: Disprove a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin Conjecture: the sum of two number is always greater than the larger number

27. Exercise: Disprove a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin Conjecture: the value of x 2 always greater than the value of x

28. Exercise: Disprove a Conjecture ERHS Math Geometry Mr. Chin-Sung Lin Conjecture: the product of two numbers is even, then the two numbers must both be even

29. Analyzing Reasoning ERHS Math Geometry Mr. Chin-Sung Lin

30. Analyzing Reasoning ERHS Math Geometry Mr. Chin-Sung Lin Use inductive reasoning to make conjectures Use deductive reasoning to show that conjectures are true or false

31. Analyzing Reasoning Example ERHS Math Geometry Mr. Chin-Sung Lin What conclusion can you make about the product of an even integer and any other integer 2 * 5 = 10(-4) * (-7) = 28 2 * 6 = 126 * 15 = 90 use inductive reasoning to make a conjecture

32. Analyzing Reasoning Example ERHS Math Geometry Mr. Chin-Sung Lin What conclusion can you make about the product of an even integer and any other integer 2 * 5 = 10(-4) * (-7) = 28 2 * 6 = 126 * 15 = 90 use inductive reasoning to make a conjecture Conjecture: Even integer * Any integer = Even integer

33. Analyzing Reasoning Example ERHS Math Geometry Mr. Chin-Sung Lin Use deductive reasoning to show that a conjecture is true Conjecture: Even integer * Any integer = Even integer Let n and m be any integer 2n is an even integer since any integer multiplied by 2 is even (2n)m represents the product of an even interger and any integer (2n)m = 2(nm) is the product of 2 and an integer nm. So, 2nm is an even integer

34. Deductive Reasoning ERHS Math Geometry Mr. Chin-Sung Lin Deductive reasoning, or deduction, is using facts, definitions, accepted properties, and the laws of logic to form a logical argument While inductive reasoning is using specific examples and patterns to form a conjecture

35. Definitions as Biconditionals ERHS Math Geometry Mr. Chin-Sung Lin

36. Definitions as Biconditionals ERHS Math Geometry Mr. Chin-Sung Lin Right angles are angles with measure of 90 Angles with measure of 90 are right angles When a conditional and its converse are both true:

37. Definitions as Biconditionals ERHS Math Geometry Mr. Chin-Sung Lin Right angles are angles with measure of 90 If angles are right angles, then their measure is 90 p  q (T) Angles with measure of 90 are right angles When a conditional and its converse are both true:

38. Definitions as Biconditionals ERHS Math Geometry Mr. Chin-Sung Lin Right angles are angles with measure of 90 If angles are right angles, then their measure is 90 p  q (T) Angles with measure of 90 are right angles If measure of angles is 90, then their are right angles q  p (T) When a conditional and its converse are both true:

39. Definitions as Biconditionals ERHS Math Geometry Mr. Chin-Sung Lin Right angles are angles with measure of 90 If angles are right angles, then their measure is 90 p  q (T) Angles with measure of 90 are right angles If measure of angles is 90, then their are right angles q  p (T) When a conditional and its converse are both true: Angles are right angles if and only if their measure is 90 q  p (T)

40. Deductive Reasoning ERHS Math Geometry Mr. Chin-Sung Lin

41. Proofs ERHS Math Geometry Mr. Chin-Sung Lin A proof is a valid argument that establishes the truth of a statement Proofs are based on a series of statements that are assume to be true Definitions are true statements and are used in geometric proofs Deductive reasoning uses the laws of logic to link together true statements to arrive at a true conclusion

42. Proofs of Euclidean Geometry ERHS Math Geometry Mr. Chin-Sung Lin given: The information known to be true prove: Statements and conclusion to be proved two-column proof: In the left column, we write statements that we known to be true In the right column, we write the reasons why each statement is true * The laws of logic are used to deduce the conclusion but the laws are not listed among the reasons

43. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: In ΔABC, AB  BC Prove: ΔABC is a right triangle Proof: StatementsReasons 1.AB  BC 1. Given.

44. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: In ΔABC, AB  BC Prove: ΔABC is a right triangle Proof: StatementsReasons 1.AB  BC 1. Given. 2.ABC is a right angle.2. If two lines are perpendicular, then they intersect to form right angles.

45. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: In ΔABC, AB  BC Prove: ΔABC is a right triangle Proof: StatementsReasons 1.AB  BC 1. Given. 2.ABC is a right angle.2. If two lines are perpendicular, then they intersect to form right angles. 3.ΔABC is a right triangle. 3. If a triangle has a right angle then it is a right triangle.

46. Paragraph Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: In ΔABC, AB  BC Prove: ΔABC is a right triangle Proof: We are given that AB  BC. If two lines are perpendicular, then they intersect to form right angles. Therefore, ABC is a right angle. A right triangle is a triangle that has a right angle. Since ABC is an angle of ΔABC, ΔABC is a right triangle.

47. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: StatementsReasons

48. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: StatementsReasons 1.BD is the bisector of ABC. 1. Given.

49. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: StatementsReasons 1.BD is the bisector of ABC. 1. Given. 2.ABD ≅ DBC 2. The bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides the angle into two congruent angles.

50. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: StatementsReasons 1.BD is the bisector of ABC. 1. Given. 2.ABD ≅ DBC 2. The bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides the angle into two congruent angles. 3.mABD = mDBC 3. Congruent angles are angles that have the same measure.

51. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AMB. Prove: AM = MB Proof: StatementsReasons

52. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AMB. Prove: AM = MB Proof: StatementsReasons 1.M is the midpoint of AMB. 1. Given.

53. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AMB. Prove: AM = MB Proof: StatementsReasons 1.M is the midpoint of AMB. 1. Given. 2.AM ≅ MB 2. The midpoint of a line segment is the point of that line segment that divides the segment into congruent segments.

54. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AMB. Prove: AM = MB Proof: StatementsReasons 1.M is the midpoint of AMB. 1. Given. 2.AM ≅ MB 2. The midpoint of a line segment is the point of that line segment that divides the segment into congruent segments. 3.AM = MB 3. Congruent segments are segments that have the same measure.

55. Direct and Indirect Proofs ERHS Math Geometry Mr. Chin-Sung Lin

56. Direct Proof ERHS Math Geometry Mr. Chin-Sung Lin A proof that starts with the given statements and uses the laws of logic to arrive at the statement to be proved is called a direct proof In most direct proofs we use definitions together with the Law of Detachment to arrive at the desired conclusion All of the proofs we have learned so far are direct proofs

57. Indirect Proof ERHS Math Geometry Mr. Chin-Sung Lin A proof that starts with the negation of the statement to be proved and uses the laws of logic to show that it is false is called an indirect proof or a proof by contradiction An indirect proof works because the negation of the statement to be proved is false, then we can conclude that the statement is true

58. Indirect Proof ERHS Math Geometry Mr. Chin-Sung Lin Let p be the given and q be the conclusion 1.Assume that the negation of the conclusion (~q) is true 2.Use this assumption (~q is true) to arrive at a statement that contradicts the given statement (p) or a true statement derived from the given statement 3.Since the assumption leads to a condiction, it (~q)must be false. The negation of the assumption (q), the desired conclusion, must be true

59. Direct Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons

60. Direct Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons 1.mCDE ≠ 90 1. Given.

61. Direct Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons 1.mCDE ≠ 90 1. Given. 2.CDE is not a right angle. 2. If the degree measure of an angle is not 90, then it is not a right angle.

62. Direct Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons 1.mCDE ≠ 90 1. Given. 2.CDE is not a right angle. 2. If the degree measure of an angle is not 90, then it is not a right angle. 3.CD is not perpendicular to 3. If two intersecting lines do not form DE right angles, then they are not perpendicular.

63. Indirect Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons

64. Indirect Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons 1.CD is perpendicular to DE 1. Assumption.

65. Indirect Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons 1.CD is perpendicular to DE 1. Assumption. 2.CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles.

66. Indirect Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons 1.CD is perpendicular to DE 1. Assumption. 2.CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles. 3.mCDE = 90. 3. If an angle is a right angle, then its degree measure is 90

67. Indirect Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons 1.CD is perpendicular to DE 1. Assumption. 2.CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles. 3.mCDE = 90. 3. If an angle is a right angle, then its degree measure is 90 4.mCDE ≠ 90 4. Given

68. Indirect Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: m  CDE ≠ 90 Prove: CD is not perpendicular to DE Proof: StatementsReasons 1.CD is perpendicular to DE 1. Assumption. 2.CDE is a right angle. 2. If two intersecting lines are perpendicular, then they form right angles. 3.mCDE = 90. 3. If an angle is a right angle, then its degree measure is 90 4.mCDE ≠ 90 4. Given 5.CD is not perpendicular to 5. Contradiction in 3 and 4. Therefore, DE the assumption is false and its negation is true.

69. Postulates, Theorems, and Proof ERHS Math Geometry Mr. Chin-Sung Lin

70. Postulate (or Axiom) ERHS Math Geometry Mr. Chin-Sung Lin A postulate (or axiom) is a statement whose truth is accepted without proof

71. Theorem ERHS Math Geometry Mr. Chin-Sung Lin A theorem is a statement that is proved by deductive reasoning

72. Theorems and Geometry ERHS Math Geometry Mr. Chin-Sung Lin undefined terms defined terms postulatestheorems applications

73. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Basic Properties of Equality Reflexive Property Symmetric Property Transitive Property Substitution Postulate Partition Postulate

74. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Addition Postulate Subtraction Postulate Multiplication Postulate Division Postulate Power Postulate Roots Postulate

75. Reflexive Property of Equality ERHS Math Geometry Mr. Chin-Sung Lin A quantity is equal to itself a = a Algebraic example: x = x

76. Reflexive Property of Equality ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: The length of a segment is equal to itself AB = AB AB

77. Symmetric Property of Equality ERHS Math Geometry Mr. Chin-Sung Lin An equality may be expressed in either order If a = b, then b = a Algebraic example: x = 5 then 5 = x

78. Symmetric Property of Equality ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If the length of AB is equal to the length of CD, then the length of CD is equal to the length of AB AB = CD then CD = AB AB CD

79. Transitive Property of Equality ERHS Math Geometry Mr. Chin-Sung Lin Quantities equal to the same quantity are equal to each other If a = b and b = c, then a = c Algebraic example: x = yandy = 4 then x = 4

80. Transitive Property of Equality ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If the lengths of segments are equal to the length of the same segment, they are equal to each other AB = EF and EF = CD then AB = CD AB EF CD

81. Substitution Postulate ERHS Math Geometry Mr. Chin-Sung Lin A quantity may be substituted for its equal in any statement of equality Algebraic example: x + y = 10 and y = 4x then x + 4x = 10

82. Substitution Postulate ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If the length of a segment is equal to the length of another segment, it can be substituted by that one in any statement of equality AB = XY and AB + BC = 10 then XY + BC = 10 ACB XY

83. Partition Postulate ERHS Math Geometry Mr. Chin-Sung Lin A whole is equal to the sum of all its parts A segment is congruent to the sum of its parts An angle is congruent to the sum of its parts Algebraic example: 2x + 3x = 5x

84. Partition Postulate ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: The sum of all the parts of a segment is congruent to the whole segment AB + BC = AC ACB

85. Addition Postulate ERHS Math Geometry Mr. Chin-Sung Lin If equal quantities are added to equal quantities, the sums are equal If congruent segments are added to congruent segments, the sums are congruent If congruent angles are added to congruent angles, the sums are congruent If a = b and c = d, then a + c = b + d Algebraic example: x - 5 = 10thenx = 15

86. Addition Postulate ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If the length of a segment is added to two equal-length segments, the sums are equal AB ≅ CD and BC ≅ BC then AB + BC ≅ CD + BC ACBD

87. Subtraction Postulate ERHS Math Geometry Mr. Chin-Sung Lin If equal quantities are subtracted from equal quantities, the differences are equal If congruent segments are subtracted to congruent segments, the differences are congruent If congruent angles are subtracted to congruent angles, the differences are congruent If a = b and c = d, then a - c = b - d Algebraic example: x + 5 = 10thenx = 5

88. Subtraction Postulate ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If a segment is subtracted from two congruent segments, the differences are congruent AC ≅ BD and BC ≅ BC then AC - BC ≅ BD - BC ACBD

89. Multiplication Postulate ERHS Math Geometry Mr. Chin-Sung Lin If equal quantities are multiplied by equal quantities, the products are equal Doubles of equal quantities are equal If a = b, and c = d, then ac = bd Algebraic example: x = 10 then 2x = 20

90. Multiplication Postulate ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If the lengths of two segments are equal, their like multiples are equal AO = CP then 2AO = 2CP AOB CPD

91. Division Postulate ERHS Math Geometry Mr. Chin-Sung Lin If equal quantities are divided by equal nonzero quantities, the quotients are equal Halves of equal quantities are equal If a = b, and c = d, then a / c = b / d (c ≠ 0 and d ≠ 0) Algebraic example: 2x = 10 then x = 5

92. Division Postulate ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If the lengths of two segments are congruent, their like divisions are congruent AB = CD then ½ AB = ½ CD AOB CPD

93. Powers Postulate ERHS Math Geometry Mr. Chin-Sung Lin The squares of equal quantities are equal If a = b, and a 2 = b 2 Algebraic example: x = 10 then x 2 = 100

94. Powers Postulate ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If the lengths of two hypotenuses are equal, their powers are equal AB = XY then AB 2 = XY 2 A B C X Y Z

95. Root Postulate ERHS Math Geometry Mr. Chin-Sung Lin Positive square roots of positive equal quantities are equal If a = b, and a > 0, then √a = √b Algebraic example: x = 100 then √x = 10

96. Root Postulate ERHS Math Geometry Mr. Chin-Sung Lin Geometric example: If the squares of the lengths of two hypotenuses are equal, their square roots are equal AB 2 = XY 2 then AB = XY A B C X Y Z

97. Identify Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin

98. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Which postulate tells us that if the measures of two angles are equal to a third angle’s, then they are equal to each other?

99. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Which postulate tells us that if the measures of two angles are equal to a third angle’s, then they are equal to each other? Transitive Property

100. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Which postulate tells us that the measure of an angle is equal to itself?

101. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Which postulate tells us that the measure of an angle is equal to itself? Reflexive Property

102. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If BD is equal to AE, and AE is equal to EC, how do we know that BD is equal to EC?

103. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If BD is equal to AE, and AE is equal to EC, how do we know that BD is equal to EC? Transitive Property

104. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin How do we know that  BAF +  FAC is equal to  BAC? B A C F

105. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin How do we know that  BAF +  FAC is equal to  BAC? B A C F Partition Postulate

106. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If BA is equal to FA, and EC is equal to AD, how do we know BA / EC = FA / AD?

107. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If BA is equal to FA, and EC is equal to AD, how do we know BA / EC = FA / AD? Division Postulate

108. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If AF is equal to AC, how do we know that AF - BD = AC - BD?

109. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If AF is equal to AC, how do we know that AF - BD = AC - BD? Subtraction Postulate

110. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If segment AB is congruent to segment XY and segment BC is congruent to segment YZ, how do we know that AB + BC = XY + YZ

111. Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Addition Postulate If segment AB is congruent to segment XY and segment BC is congruent to segment YZ, how do we know that AB + BC = XY + YZ

112. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin

113. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If AB  BC and LM  MN, prove mABC = mLMN

114. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If AB  BC and LM  MN, prove mABC = mLMN Given: AB  BC and LM  MN Prove: mABC = mLMN A B C L M N

115. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB  BC and LM  MN Prove: mABC = mLMN Proof: StatementsReasons

116. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB  BC and LM  MN Prove: mABC = mLMN Proof: StatementsReasons 1.AB  BC and LM  MN 1. Given.

117. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB  BC and LM  MN Prove: mABC = mLMN Proof: StatementsReasons 1.AB  BC and LM  MN 1. Given. 2.ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right angles.

118. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB  BC and LM  MN Prove: mABC = mLMN Proof: StatementsReasons 1.AB  BC and LM  MN 1. Given. 2.ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right angles. 3. mABC = 90, mLMN = 90 3. A right angle is an angle whose degree measure is 90

119. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB  BC and LM  MN Prove: mABC = mLMN Proof: StatementsReasons 1.AB  BC and LM  MN 1. Given. 2.ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right angles. 3. mABC = 90, mLMN = 90 3. A right angle is an angle whose degree measure is 90 4.90 = mLMN 4. Symmetric property of equality

120. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB  BC and LM  MN Prove: mABC = mLMN Proof: StatementsReasons 1.AB  BC and LM  MN 1. Given. 2.ABC and LMN are right 2. Perpendicular lines are two angles. lines intersecting to form right angles. 3. mABC = 90, mLMN = 90 3. A right angle is an angle whose degree measure is 90 4. 90 = mLMN 4. Symmetric property of equality 5. mABC = mLMN 5. Transitive property of equality

121. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If AB = 2 CD, and CD = XY, prove AB = 2 XY

122. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If AB = 2 CD, and CD = XY, prove AB = 2 XY Given: AB = 2 CD, and CD = XY Prove: AB = 2 XY AC B D XY

123. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB = 2 CD, and CD = XY Prove: AB = 2 XY Proof: StatementsReasons

124. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB = 2 CD, and CD = XY Prove: AB = 2 XY Proof: StatementsReasons 1.AB = 2 CD, and CD = XY 1. Given.

125. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: AB = 2 CD, and CD = XY Prove: AB = 2 XY Proof: StatementsReasons 1.AB = 2 CD, and CD = XY 1. Given. 2.AB = 2 XY 2. Substitution postulate

126. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If ABCD are collinear, AB = CD, prove AC = BD

127. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin If ABCD are collinear, AB = CD, prove AC = BD Given: ABCD and AB = CD Prove: AC = BD ACBD

128. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: ABCD and AB = CD Prove: AC = BD Proof: StatementsReasons

129. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: ABCD and AB = CD Prove: AC = BD Proof: StatementsReasons 1.AB = CD 1. Given.

130. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: ABCD and AB = CD Prove: AC = BD Proof: StatementsReasons 1.AB = CD 1. Given. 2.BC = BC 2. Reflexive property of equality

131. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: ABCD and AB = CD Prove: AC = BD Proof: StatementsReasons 1.AB = CD 1. Given. 2.BC = BC 2. Reflexive property of equality 3.AB + BC = CD + BC 3. Addition postulate

132. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: ABCD and AB = CD Prove: AC = BD Proof: StatementsReasons 1.AB = CD 1. Given. 2.BC = BC 2. Reflexive property of equality 3.AB + BC = CD + BC 3. Addition postulate 4.AB + BC = AC 4. Partition postulate CD + BC = BD

133. Apply Postulates for Proofs ERHS Math Geometry Mr. Chin-Sung Lin Given: ABCD and AB = CD Prove: AC = BD Proof: StatementsReasons 1.AB = CD 1. Given. 2.BC = BC 2. Reflexive property of equality 3.AB + BC = CD + BC 3. Addition postulate 4.AB + BC = AC 4. Partition postulate CD + BC = BD 5. AC = BD 5. Substitution postulate

134. Q & A ERHS Math Geometry Mr. Chin-Sung Lin

135. The End ERHS Math Geometry Mr. Chin-Sung Lin