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Geometry Section 2-6

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.

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Section 2-6 Algebraic Proof Wednesday, November 6, 13
Essential Questions How do you use Algebra to write two-column proofs? How do you use properties of equality to write geometric proofs? Wednesday, November 6, 13
Vocabulary 1. Algebraic Proof: 2. Two-column Proof: 3. Formal Proof: Wednesday, November 6, 13
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: Wednesday, November 6, 13
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: Wednesday, November 6, 13
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 Wednesday, November 6, 13
Properties of Real Numbers Wednesday, November 6, 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
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
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
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
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
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
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
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
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
Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Wednesday, November 6, 13
Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Wednesday, November 6, 13 Given
Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 10 − 6a − 4a − 28 = 92 Wednesday, November 6, 13 Given
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 Wednesday, November 6, 13
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 Wednesday, November 6, 13
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 Wednesday, November 6, 13
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 Wednesday, November 6, 13
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 Wednesday, November 6, 13 Addition Property
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 Wednesday, November 6, 13 Addition Property
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
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
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 Wednesday, November 6, 13 Addition Property
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 Wednesday, November 6, 13 Addition Property
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 Wednesday, November 6, 13 Addition Property
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. Wednesday, November 6, 13
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 Wednesday, November 6, 13 Reasons
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. Wednesday, November 6, 13 Reasons
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 Wednesday, November 6, 13 Reasons
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 Wednesday, November 6, 13 Reasons Given
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. Wednesday, November 6, 13 Reasons Given
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 Wednesday, November 6, 13 Reasons Given
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 Wednesday, November 6, 13 Reasons Given Subtraction Property
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. Wednesday, November 6, 13 Reasons Given Subtraction Property
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 Wednesday, November 6, 13 Reasons Given Subtraction Property
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 Wednesday, November 6, 13 Reasons Given Subtraction Property Division Property
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. Wednesday, November 6, 13 Reasons Given Subtraction Property Division Property
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. Wednesday, November 6, 13 d −5 t= 20 Reasons Given Subtraction Property Division Property
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. Wednesday, November 6, 13 d −5 t= 20 Reasons Given Subtraction Property Division Property Symmetric Property
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
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 Wednesday, November 6, 13 Reasons
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. Wednesday, November 6, 13 Reasons
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° Wednesday, November 6, 13 Reasons
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° Wednesday, November 6, 13 Reasons Given
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. Wednesday, November 6, 13 Reasons Given
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. Wednesday, November 6, 13 m∠A = m∠B Reasons Given
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. Wednesday, November 6, 13 m∠A = m∠B Reasons Given Def. of ≅ angles
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. Wednesday, November 6, 13 m∠A = m∠B Reasons Given Def. of ≅ angles
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 Wednesday, November 6, 13 Reasons Given Def. of ≅ angles
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 Wednesday, November 6, 13
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. Wednesday, November 6, 13
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
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
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. Wednesday, November 6, 13
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
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 Wednesday, November 6, 13
Problem Set Wednesday, November 6, 13
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 Wednesday, November 6, 13

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Geometry Section 2-6

  • 1.
  • 2.
    Essential Questions How doyou use Algebra to write two-column proofs? How do you use properties of equality to write geometric proofs? Wednesday, November 6, 13
  • 3.
    Vocabulary 1. Algebraic Proof: 2.Two-column Proof: 3. Formal Proof: Wednesday, November 6, 13
  • 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: Wednesday, November 6, 13
  • 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: Wednesday, November 6, 13
  • 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 Wednesday, November 6, 13
  • 7.
    Properties of RealNumbers Wednesday, November 6, 13
  • 8.
    Properties of RealNumbers 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 RealNumbers 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 RealNumbers 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 RealNumbers 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 RealNumbers 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 RealNumbers 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 RealNumbers 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 RealNumbers 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 RealNumbers 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 theequation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Wednesday, November 6, 13
  • 18.
    Example 1 Solve theequation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Wednesday, November 6, 13 Given
  • 19.
    Example 1 Solve theequation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 10 − 6a − 4a − 28 = 92 Wednesday, November 6, 13 Given
  • 20.
    Example 1 Solve theequation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property Wednesday, November 6, 13
  • 21.
    Example 1 Solve theequation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 Wednesday, November 6, 13
  • 22.
    Example 1 Solve theequation. 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 Wednesday, November 6, 13
  • 23.
    Example 1 Solve theequation. 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 Wednesday, November 6, 13
  • 24.
    Example 1 Solve theequation. 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 Wednesday, November 6, 13 Addition Property
  • 25.
    Example 1 Solve theequation. 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 Wednesday, November 6, 13 Addition Property
  • 26.
    Example 1 Solve theequation. 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 theequation. 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 theequation. 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 Wednesday, November 6, 13 Addition Property
  • 29.
    Example 1 Solve theequation. 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 Wednesday, November 6, 13 Addition Property
  • 30.
    Example 1 Solve theequation. 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 Wednesday, November 6, 13 Addition Property
  • 31.
    Example 2 If thedistance 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. Wednesday, November 6, 13
  • 32.
    Example 2 If thedistance 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 Wednesday, November 6, 13 Reasons
  • 33.
    Example 2 If thedistance 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. Wednesday, November 6, 13 Reasons
  • 34.
    Example 2 If thedistance 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 Wednesday, November 6, 13 Reasons
  • 35.
    Example 2 If thedistance 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 Wednesday, November 6, 13 Reasons Given
  • 36.
    Example 2 If thedistance 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. Wednesday, November 6, 13 Reasons Given
  • 37.
    Example 2 If thedistance 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 Wednesday, November 6, 13 Reasons Given
  • 38.
    Example 2 If thedistance 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 Wednesday, November 6, 13 Reasons Given Subtraction Property
  • 39.
    Example 2 If thedistance 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. Wednesday, November 6, 13 Reasons Given Subtraction Property
  • 40.
    Example 2 If thedistance 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 Wednesday, November 6, 13 Reasons Given Subtraction Property
  • 41.
    Example 2 If thedistance 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 Wednesday, November 6, 13 Reasons Given Subtraction Property Division Property
  • 42.
    Example 2 If thedistance 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. Wednesday, November 6, 13 Reasons Given Subtraction Property Division Property
  • 43.
    Example 2 If thedistance 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. Wednesday, November 6, 13 d −5 t= 20 Reasons Given Subtraction Property Division Property
  • 44.
    Example 2 If thedistance 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. Wednesday, November 6, 13 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 Wednesday, November 6, 13 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. Wednesday, November 6, 13 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° Wednesday, November 6, 13 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° Wednesday, November 6, 13 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. Wednesday, November 6, 13 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. Wednesday, November 6, 13 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. Wednesday, November 6, 13 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. Wednesday, November 6, 13 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 Wednesday, November 6, 13 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 Wednesday, November 6, 13
  • 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. Wednesday, November 6, 13
  • 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. Wednesday, November 6, 13
  • 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 Wednesday, November 6, 13
  • 62.
  • 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 Wednesday, November 6, 13