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2.4 Reasoning with Properties from Algebra Algebraic Properties of Equality 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, then a/c = b/c (c ≠ 0) Reflexive Property For any real number a, a = a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c then a = c Substitution Property If a = b, then a can be substituted for b in any equation or expression. Handout with other properties students are expected to KNOW!!!!! Solve 5x 18 = 3x + 2 and write a reason for each step. 1) 5x 18 = 3x + 2 1) Given 2) 2x 18 = 2 2) Subtraction Property of Equality 3) 2x = 20 3) Addition 4) x = 10 4) Division Solve 55x 3(9x+ 12) = 64 and write a reason for each step. Solve for r and give a reason for each step. a = 220 (10/7)r 1 If (1/2x) 4 = 8, then x = 24. Give a reason for each step. 1) (1/2)x 4 = 8 1) 2) (1/2)x 4 + 4 = 8 + 4 2) 3) (1/2)x + 0 = 8 + 4 3) 4) (1/2)x = 8 + 4 4) 5) (1/2)x = 12 5) 6) 2 (1/2)x = 2 ∙ 12 6) 7) 1x = 2 12 7) 8) x = 2 12 8) 9) x = 24 9) You are not expected to come up with this detailed proof, but you are expected to be able to fill in the blank for this type of proof. 2 Properties of Length (Rules for Segments) and Measure (Rules for Angles) Segment Length Angle Measure Reflexive For any segment AB, For any angle A, AB = AB. m<A = m<A. (This says that a segment or an angle is always equal to itself.) Symmetric If AB = CD, then If m<A = m<B, then CD = AB. m<B = m<A. If the measure of two segments or angles are equal, then you can reverse the order around the equal sign. Transitive If AB = CD and CD = EF, If m<A = m<B and then AB = EF. m<B = m<C, then m<A = m<C. ex) 2 + 6 = 3 + 5 and 3 + 5 = 1 + 7, then 2 + 6 = 1 + 7. ex) AB = CD Give an argument that shows AC = BD. A B C D 1) AB = CD 1) Given 2) AB + BC = BC + CD 2) 3) AB + BC = AC 3) 4) BC + CD = BD 4) 5) AC = BD 5) 3 Given: m<1 + m<2 = 66 m<1 + m<2 + m<3 = 99 m<3 = m<1 m<1 = m<4 Find the m<4 and give a reason for your thinking. 1) m<1 + m<2 = 66 1) Given m<1 + m<2 + m<3 = 99 m<3 = m<1 m<1 = m<4 2) 66 + m<3 = 99 2) 3) m<3 = 33 3) 4) m<3 = m<4 4) 5) m<4 = 33 ` 5) When you are giving reasons for a logical argument, don't overlook the given information. This will always be your first step in a proof. I always list all my given in the first step. The book does not do this. They some times put given steps in the middle of an argument. Either way is acceptable. Assignment QUIZ THURSDAY 2.1 2.3 and Properties page 100: 15 20, 24, 25, 28, 32, 36, 42, 46 50 ALL 4