2.6 Algebraic Proof 101708

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2.6 Algebraic Proof 10/17/08

• Algebra is a system with sets of numbers,
  operations, and properties that allow you to
  perform algebraic operations.

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Properties of Equality for Real Numbers

• Reflexive Property a a
• Symmetric Property if a b, then b a
• Transitive Property if a b and b c, then a
  c
• Addition and Subtraction Property if a b,
  then a c b c, and a c b c
• Multiplication and Division Propertyif a b,
  then a c b c, and if c ? 0, then a/c b/c
• Substitution PropertyFor all numbers a and b,
  if a b, then a may be replaced by b in any
  equation or expression.
• Distributive Property a (b c) ab ac

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• The properties of equality can be used to justify
  each step when solving an equation.
• A group of algebraic steps used to solve problems
  form a deductive argument.

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Verify Algebraic Relationships
Solve 3(x 2) 42

• Algebraic Steps
• 3(x 2) 42
• 3x 6 42
• 3x 6 6 42 6
• 3x 48
• x 16

• Properties
• Original Equation
• Distributive Property
• Addition Property
• Substitution
• Division
• Substitution

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TwoColumn Proofs

• A twocolumn proof, or formal proof, contains
  statements and reasons organized in two columns.
• In a two-column proof, each step is called a
  statement.
• The properties that justify each step are called
  reasons.

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Write a TwoColumn proof

• Statements
• 1.
• 2.
• 3x 5 1
• 3x 5 5 1 5
• 3x 6
• x 2

• Reasons
• 1. Given
• 2. Distributive Property
• Substitution
• Addition
• Substitution
• Division
• Substitution

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Geometric Proof
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Geometric Proof

• On a clock, the angle formed by the hands at 200
  is a 60 angle. If the angle formed at 200 is
  congruent to the angle formed at 1000, prove
  that the angle at 1000 is a 60 angle.

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Geometric Proof Cont.
GivenProve

• Statements
• 1.
• 2.
• 3.
• 4.

• Reasons
• Given
• Definition of congruent angles.
• Substitution
• Symmetric Property

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Extra Practice!!!

• Classwork Study Guide 2.6
• Homework Practice 2.6