Section 2-4: Reasoning in Algebra TPI 32A: apply reflective, transitive, or symmetric prooperties of equality or congruence Objectives: Connect reasoning in algebra and geometry Justify steps in deductive reasoning In geometry postulates, definitions, & properties are accepted as true you use deductive reasoning to prove other statements We will look at some basic properties used to justify statements….. ….. which leads to writing proofs.

Properties of Equality Addition Property of Equality If a = b, then a + c = b + c Add same amount to both sides of an equation. Subtraction Property of Equality If a = b, then a - c = b - c Subtract same amount to both sides of an equation. Multiplication Property of Equality If a = b, then a ∙ c = b ∙ c Multiply both sides of an equation by the same amount. Division Property of Equality If a = b and c  0, then Divide both sides of an equation by the same amount.

Properties of Equality (cont) Reflective Property of Equality a = a Ex: 5 = 5 Symmetric Property of Equality If a = b, then b = a Ex: 3 = 2 + 1 and 2 + 1 = 3 are the same. Transitive Property of Equality If a = b and b = c, then a = c. EX: If 3 + 4 = 7 and 5 + 2 = 7, then 3 + 4 = 5 + 2. Substitution Property of Equality If a = b , then b can replace a in any expression. Ex: a = 3; If a = b, then 3 = 3. Distributive Property a(b + c) = ab + ac Ex: 3(x + 3) = 3x + 9

Using Properties to Justify Steps in Solving Equations Algebra Solve for x and justify each step. Given: m AOC = 139 m AOC = 139 Given m AOB + m BOC = m AOC Angle Addition Postulate x + 2x + 10 = 139 Substitution Property Simplify 3x + 10 = 139 3x = 129 Subtraction Property of Equality x = 43 Division Property of Equality

Using Properties to Justify Steps in Solving Equations Solve for x and justify each step. Given: LM bisects KLN LM bisects KLN Given MLN = KLM 4x = 2x + 40 2x = 40 x = 20 Def of Angle Bisector Substitution Property Subtraction Property of Equality Division Property of Equality

Using Properties to Justify Steps in Solving Equations Solve for y and justify each step Given: AC = 21 AC = 21 Given AB + BC = AC Segment Addition Postulate 2y + 3y - 9 = 21 Substitution Property Simplify 5y – 9 = 21 5y = 30 Addition Property of Equality y = 6 Division Property of Equality Find AB and BC by substituting y = 6 into the expressions.

Properties of Congruence The Reflective, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence that can be used to justify statements. Reflective Property of Congruence AB  AB A  A Symmetric Property of Congruence If AB  CD, then CD  AB. If A  B, then B  A Transitive Property of Congruence If AB  CD and AB  EF, then CD  EF. If A  B and B  C, then A  C.

Using Properties of Equality and Congruence Name the property of congruence or equality the justifies each statement. a. K  K Reflective Property of  b. If 2x – 8 = 10, then 2x = 18 Addition Property of Equality c. If RS  TW and TW  PQ, then RS  PQ. Transitive Property of  d. If m A = mB, then m B = mA Symmetric Property of Equality

Use what you know about transitive properties to complete the following: The Transitive Property of Falling Dominoes: If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino _______ to fall. C