2 5 Algebraic Proof Objectives Review properties of

2 -5 Algebraic Proof Objectives Review properties of equality and use them to write algebraic proofs. Holt Mc. Dougal Geometry

2 -5 Algebraic Proof A proof uses logic, definitions, properties, and previously proven statements to “prove” that a conclusion is true. An important part of writing a proof is giving justifications to show that every step is valid. Possible ways to “prove” or justify each step: 1. A definition 2. A postulate 3. A property 4. Something given in the problem Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Remember! The Distributive Property states that a(b + c) = ab + ac. Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Example 1: Solving an Equation in Algebra Solve the equation 4 m – 8 = – 12. Write a justification for each step. 4 m – 8 = – 12 +8 +8 4 m = – 4 m = – 1 Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Check It Out! Example 1 Solve the equation for each step. t = – 14 Holt Mc. Dougal Geometry . Write a justification

2 -5 Algebraic Proof Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry. Helpful Hint A B AB represents the length AB, so you can think of AB as a variable representing a number. Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Example 2: Solving an Equation in Geometry Write a justification for each step. NO = NM + MO 4 x – 4 = 2 x + (3 x – 9) 4 x – 4 = 5 x – 9 – 4 = x – 9 5=x Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Check It Out! Example 2 Write a justification for each step. m ABC = m ABD + m DBC 8 x° = (3 x + 5)° + (6 x – 16)° 8 x = 9 x – 11 –x = – 11 x = 11 Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Classwork: Exit Ticket with Partner Homework: • 2. 5 Practice B Homework Handout • Must write out justification for each step to get credit!! Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Lesson Quiz: Part I Solve each equation. Write a justification for each step. 1. Given z – 5 = – 12 z = – 7 Holt Mc. Dougal Geometry Mult. Prop. of = Add. Prop. of =

2 -5 Algebraic Proof Lesson Quiz: Part II Solve each equation. Write a justification for each step. 2. 6 r – 3 = – 2(r + 1) Given 6 r – 3 = – 2 r – 2 Distrib. Prop. 8 r – 3 = – 2 8 r = 1 Add. Prop. of = Div. Prop. of = Holt Mc. Dougal Geometry

2 -5 Algebraic Proof WARM UP Give an example of the following properties: 1. Reflexive Property of Equality: 2. Symmetric Property of Equality: 3. Transitive Property of Equality: Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Objective: Identify properties of equality and congruence. Remember…. . segments with equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Remember! Numbers are equal (=) and figures are congruent ( ). Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Example 1: Identifying Property of Equality and Congruence Identify the property that justifies each statement. A. QRS Reflex. Prop. of . B. m 1 = m 2 so m 2 = m 1 Symm. Prop. of = C. AB CD and CD EF, so AB EF. Trans. Prop of D. 32° = 32° Reflex. Prop. of = Holt Mc. Dougal Geometry

2 -5 Algebraic Proof Check It Out! Example 2 Identify the property that justifies each statement. A. DE = GH, so GH = DE. Sym. Prop. of = B. 94° = 94° Reflex. Prop. of = C. 0 = a, and a = x. So 0 = x. D. A Y, so Y A Holt Mc. Dougal Geometry Trans. Prop. of = Sym. Prop. of

2 -5 Algebraic Proof Lesson Review Identify the property that justifies each statement. 1. x = y and y = z, so x = z. Trans. Prop. of = 2. DEF Reflex. Prop. of 3. AB CD, so CD AB. Holt Mc. Dougal Geometry Sym. Prop. of

2 -5 Algebraic Proof Classwork/Homework come up with a unique example of each Holt Mc. Dougal Geometry