NAME DATE PERIOD 2-6 Study Guide and Intervention

1 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, DATE PERIOD Lesson 2-6 Chapter 2 37 Glencoe GeometryAlgebraic Proof A list of algebraic steps to solve problems where each step is justified is called an algebraic proof, The table shows properties you have studied in following properties are true for any real numbers a, b, and Property of EqualityIf a = b, then a + c = b + Property of EqualityIf a = b, the a - c = b - Property of EqualityIf a = b, then a c = b Property of EqualityIf a = b and c 0, then, a c = b c.

2 Reflexive Property of Equalitya = a Symmetric Property of EqualityIf a = b and b = Property of EqualityIf a = b and b = c, then a = Property of EqualityIf a = b,then a may be replaced by b in any equation or Propertya (b + c) = ab + ac Solve 6x + 2(x - 1) = 30. Write a justification for each Steps Properties 6x + 2(x - 1) = 30 Original equation or Given 6x + 2x - 2 = 30 Distributive Property 8x - 2 = 30 Substitution Property of Equality 8x - 2 + 2 = 30 + 2 Addition Property of Equality 8x = 32 Substitution Property of Equality 8x 8 = 32 8 Division Property of Equality x = 4 Substitution Property of EqualityExercisesComplete each Guide and InterventionAlgebraic Proof2-6 Example 2.

3 Given: 4x + 8 = x + 2 Prove: x = -2 Proof: Statements Reasonsa. 4x + 8 = x + 2 a. b. 4x + 8 - x = b. x + 2 - x c. 3x + 8 = 2 c. Substitutiond. d. Subtr. e. Substitutionf. 3x 3 = -6 3 f. g. g. Substitution 1. Given: 4x + 6 2 = 9 Prove: x = 3 Proof: Statements Reasonsa. 4x + 6 2 = 9 a. b. ( 4x + 6 2 ) = 2(9) b. Mult. 4x + 6 = 18 c. d. 4x + 6 - 6 = 18 - 6 d. e. 4x = e. Substitutionf. 4x 4 = f. Div. g. = 3 GivenSubtr.

4 Prop. 3x + 8 - 8 = 2 - 83x = -6 Div. = -2 12 4 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, DATE PERIOD Chapter 2 38 Glencoe GeometryGeometric Proof geometry deals with numbers as measures, so geometric proofs use properties of numbers. Here are some of the algebraic properties used in proofs. Write a two-column proof to verify this conjecture. Given: m 1 = m 2, m 2 = m 3 Prove: m 1 = m 3 Proof:ExercisesState the property that justifies each statement. 1. If m 1 = m 2, then m 2 = m 1. 2. If m 1 = 90 and m 2 = m 1, then m 2 = 90.

5 3. If AB = RS and RS = WY, then AB = WY. 4. If AB = CD, then 1 2 AB = 1 2 CD. 5. If m 1 + m 2 = 110 and m 2 = m 3, then m 1 + m 3 = 110. 6. RS = RS 7. If AB = RS and TU = WY, then AB + TU = RS + WY. 8. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. 9. If the formula for the area of a triangle is A = 1 2 bh, then bh is equal to 2 times the area of the triangle. Write a two-column proof to verify this Guide and Intervention (continued)Algebraic Proof2-6 PropertySegmentsAnglesReflexiveAB = ABm 1 = m 1 SymmetricIf AB = CD, then CD = m 1 = m 2, then m 2 = m AB = CD and CD = EF, then AB = m 1 = m 2 and m 2 = m 3, then m 1= m m 1 = m 22.

6 M 2 = m 33. m 1 = m 31. Given2. Given3. Transitive Property of EqualitySym. Prop. Prop. Mult. : A = 1 2 bhProve: 2A = bhProof:StatementsReasons1. A = 1 2 bh2. 2A = 2 ( 1 2 bh) 3. 2A = bh1. Given2. Mult. Property3. Subs.