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2.5 and 2.6 Properties of Equality and Congruence Objective: You will use deductive reasoning to: – – Write proofs using geometric theorem To use algebraic properties in logical arguments. Algebraic Properties Substitution Property: If a = b, then a can be substituted for b in an equation or expression. Distributive Property: a(b + c) = ab + ac, where a, b, and c are real numbers. Algebraic Properties Addition Property: If a = b, then a + c = b + c. Subtraction Property: If a = b, then a – c = b – c. Multiplication Property: If a = b, then ac = bc. Division Property: If a = b and c = 0, then a/c = b/c. Example 1: Write a two-column proof to solve the equation. 3x + 2 = 8 Statements 1. 2. 3. 4. 5. 3x + 2 = 8 3x + 2 – 2 = 8 – 2 3x = 6 3x ÷ 3 = 6 ÷ 3 x=2 Reasons 1. Given 2. Subtraction Prop 3. Simplify 4. Division Prop 5. Simplify Example 2: Write a two-column proof to solve the equation. Statements Reasons Given 6. 4x + 9 = 16 – 3x 4x + 9 + 3x = 16 – 3x + 3x 7x + 9 = 16 7x + 9 – 9 = 16 – 9 7x = 7 7x ÷ 7 = 7 ÷ 7 7. x=1 Simplify 1. 2. 3. 4. 5. Addition Prop Simplify Subtraction Prop Simplify Division Prop Example 3: Write a two-column proof to solve the equation. 2(-x – 5) = 12 Statements 1. 2. 3. 4. 5. 6. 2(-x – 5) = 12 -2x – 10 = 12 -2x – 10 + 10 = 12 + 10 -2x = 22 -2x ÷ -2 = 22 ÷ -2 Prop x = -11 Reasons Given Distributive Prop Addition Prop Simplify Division Simplify Reflexive Property Equality AB AB mA mA Congruence AB AB A A Symmetric Property Equality: If AB CD, then CD AB If mA mB, then mB mA Congruence: If AB CD, then CD AB If A B, then B A Transitive Property Equality: If AB CD & CD EF , then AB EF If mA mB & mB mC , then mA mC Congruence: If AB CD & CD EF , then AB EF If A B & B C , then A C Properties of Equality Addition Property: adding a number to each side of an equation Subtraction Property: subtracting a number from each side of an equation Properties of Equality Multiplication Property: multiplying by a number on each side of an equation Division Property: dividing by a number on each side of an equation Substitution Property: substituting a number for a variable in an equation to produce an equivalent equation Definitions Theorem: A true statement that follows as a result of other true statements. Two-column proof: Most commonly used. Has numbered statements and reasons that show the logical order of an argument. Use the diagram and the given information to complete the missing steps and reasons in the proof. GIVEN: LK = 5, JK = 5, JK ≅ JL PROVE: LK ≅ JL K Statements: Reasons: J 1. 2. 3. 4. 5. 6. _______________ _______________ LK = JK LK ≅ JK JK ≅ JL ________________ 1. 2. 3. 4. 5. 6. Given Given Transitive Property _______________ Given Transitive Property L