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Page 1: Warm Up. Warm Up Answers Theorem and Proof A theorem is a statement or conjecture that has been shown to be true. A theorem is a statement or conjecture

WarmWarmUpUp

Page 2: Warm Up. Warm Up Answers Theorem and Proof A theorem is a statement or conjecture that has been shown to be true. A theorem is a statement or conjecture

Warm Up AnswersWarm Up Answers

Page 3: Warm Up. Warm Up Answers Theorem and Proof A theorem is a statement or conjecture that has been shown to be true. A theorem is a statement or conjecture

Theorem and ProofTheorem and Proof

A A theoremtheorem is a statement or conjecture that has is a statement or conjecture that has been shown to be true. been shown to be true.

Theorems can be used like a definition or Theorems can be used like a definition or postulate to justify other statements are true.postulate to justify other statements are true.

A A proofproof is a logical argument in which each is a logical argument in which each statement made is supported by a statement statement made is supported by a statement that is accepted as true.that is accepted as true.

A A paragraph proofparagraph proof or or informal proofinformal proof is one is one type of proof.type of proof.

Page 4: Warm Up. Warm Up Answers Theorem and Proof A theorem is a statement or conjecture that has been shown to be true. A theorem is a statement or conjecture

2.5 Algebraic Proof2.5 Algebraic Proof

Algebraic proofs use algebra to write two-Algebraic proofs use algebra to write two-column proofs.column proofs.

Two-Column ProofsTwo-Column Proofs or or formal proofsformal proofs contains contains statements and reasons organized into 2 statements and reasons organized into 2 columns.columns. Each step is called a statement and the properties Each step is called a statement and the properties

that justify each step are called the reasons. that justify each step are called the reasons.

Page 5: Warm Up. Warm Up Answers Theorem and Proof A theorem is a statement or conjecture that has been shown to be true. A theorem is a statement or conjecture

Properties of Equality for Real Properties of Equality for Real NumbersNumbers

Reflexive PropertyReflexive Property: a = a: a = a Symmetric PropertySymmetric Property: if a = b, then b = a: if a = b, then b = a Transitive PropertyTransitive Property: if a=b, and b=c, then a=c: if a=b, and b=c, then a=c Addition Property:Addition Property: if a=b, then a+c = b+c if a=b, then a+c = b+c Subtraction PropertySubtraction Property: if a=b, then a-c = b-c: if a=b, then a-c = b-c Multiplication/DivisionMultiplication/Division: if a=b, then ac = bc: if a=b, then ac = bc Substitution PropertySubstitution Property: if a=b, then : if a=b, then aa may be may be

replaced by replaced by bb in any equation or expression in any equation or expression Distributive PropertyDistributive Property: a(b+c) = ab + ac: a(b+c) = ab + ac

Page 6: Warm Up. Warm Up Answers Theorem and Proof A theorem is a statement or conjecture that has been shown to be true. A theorem is a statement or conjecture

ExampleExample

Complete the following proof.Complete the following proof.Given: 5- ½ x = 1Given: 5- ½ x = 1Prove: 8 = xProve: 8 = xStatementsStatements ReasonsReasons1.1. 5 – ½ x = 15 – ½ x = 1 1. Given1. Given2.2. 5 – ½ x – 5 = 1 – 55 – ½ x – 5 = 1 – 5 2. ____________2. ____________3.3. - ½ x = -4- ½ x = -4 3. ____________3. ____________4.4. ______________________________ 4. Multiplication4. Multiplication5.5. x = 8x = 8 5. _____________5. _____________6.6. 8 = x8 = x 6. _____________6. _____________

Page 7: Warm Up. Warm Up Answers Theorem and Proof A theorem is a statement or conjecture that has been shown to be true. A theorem is a statement or conjecture

AnswerAnswer

Complete the following proof.Complete the following proof.Given: 5- ½ x = 1Given: 5- ½ x = 1Prove: 8 = xProve: 8 = xStatementsStatements ReasonsReasons1.1. 5 – ½ x = 15 – ½ x = 1 1. Given1. Given2.2. 5 – ½ x – 5 = 1 – 55 – ½ x – 5 = 1 – 5 2. 2. SubtractionSubtraction3.3. - ½ x = -4- ½ x = -4 3. 3. SubstitutionSubstitution4.4. -2( ½ x) = -2(-4)-2( ½ x) = -2(-4) 4. Multiplication4. Multiplication5.5. x = 8x = 8 5. 5. SubstitutionSubstitution6.6. 8 = x8 = x 6. 6. Symmetric Prop.Symmetric Prop.


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