1. Warm Up

2. Warm Up Answers

3. Theorem and Proof A theorem is a statement or conjecture that has been shown to be true. A theorem is a statement or conjecture that has been shown to be true. Theorems can be used like a definition or postulate to justify other statements are true. Theorems can be used like a definition or postulate to justify other statements are true. A proof is a logical argument in which each statement made is supported by a statement that is accepted as true. A proof is a logical argument in which each statement made is supported by a statement that is accepted as true. A paragraph proof or informal proof is one type of proof. A paragraph proof or informal proof is one type of proof.

4. 2.5 Algebraic Proof Algebraic proofs use algebra to write two- column proofs. Algebraic proofs use algebra to write two- column proofs. Two-Column Proofs or formal proofs contains statements and reasons organized into 2 columns. Two-Column Proofs or formal proofs contains statements and reasons organized into 2 columns. Each step is called a statement and the properties that justify each step are called the reasons. Each step is called a statement and the properties that justify each step are called the reasons.

5. Properties of Equality for Real Numbers Reflexive Property: a = a Reflexive Property: a = a Symmetric Property: if a = b, then b = a Symmetric Property: if a = b, then b = a Transitive Property: if a=b, and b=c, then a=c Transitive Property: if a=b, and b=c, then a=c Addition Property: if a=b, then a+c = b+c Addition Property: if a=b, then a+c = b+c Subtraction Property: if a=b, then a-c = b-c Subtraction Property: if a=b, then a-c = b-c Multiplication/Division: if a=b, then ac = bc Multiplication/Division: if a=b, then ac = bc Substitution Property: if a=b, then a may be replaced by b in any equation or expression Substitution Property: if a=b, then a may be replaced by b in any equation or expression Distributive Property: a(b+c) = ab + ac Distributive Property: a(b+c) = ab + ac

6. Example Complete the following proof. Given: 5- ½ x = 1 Prove: 8 = x StatementsReasons 1. 5 – ½ x = 11. Given 2. 5 – ½ x – 5 = 1 – 52. ____________ 3. - ½ x = -43. ____________ 4. _______________4. Multiplication 5. x = 85. _____________ 6. 8 = x6. _____________

7. Answer Complete the following proof. Given: 5- ½ x = 1 Prove: 8 = x StatementsReasons 1. 5 – ½ x = 11. Given 2. 5 – ½ x – 5 = 1 – 52. Subtraction 3. - ½ x = -43. Substitution 4. -2( ½ x) = -2(-4)4. Multiplication 5. x = 85. Substitution 6. 8 = x6. Symmetric Prop.