2.6 algebraic proofs. September 13, algebraic proofs ink.notebook. Page 71. Page 72 Page 70. Page 73. Standards. 2.

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1 2.6 algebraic proofs ink.notebook September 13, 2017 Page 71 Page 72 Page algebraic proofs Page 73 Lesson Objectives Standards Lesson Notes 2.6 Algebraic Proofs Press the tabs to view details. 1

2 Lesson Objectives Standards Lesson Notes Lesson Objectives Standards Lesson Notes After this lesson, you should be able to successfully apply algebraic properties to write proofs. G.CO.9 Prove theorems about lines and angles. Press the tabs to view details. Press the tabs to view details. Addition of Equality Subtraction 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 algebra. If a = b, then a + c = b + c Multiplication If a = b, then ac = bc Reflexive If a = b, then a - c = b - c Division of Equality If a = b and c = 0, then a c = b c a = a If a = b, then b = a Transitive If a = b and b = c, then a = c Substitution If a = b, then a may be replaced by b in any equation 2

3 2.6 algebraic proofs ink.notebook September 13, 2017 Example 1: Solve 2x + 3 = 9 x. Write a reason for each step. Distributive : a(b + c) = ab + bc Equation Explanation 2x + 3 = 9 x Rewrite the original equation 2x = 9 x + Add to each side Reason Given Addition Substitution + 3 = Combine like terms = Subtract from each side x = Divide each side by Subtraction Division Example 2: Solve 4(6x + 2) = 64. Write a reason for each step. Equation Explanation 4(6x + 2) = 64 Rewrite the original equation = 64 Multiply = Add to each side x = Divide each side by Reason Given Addition Division *Do not have to show the substitution step to show you simplified!! 3

two columns Each step is called Statement Properties that justify each step are called Reason Geometric Proof Geometry deals with numbers

Segments Angles Reflexive AB = AB m<1 = m<1 Symmetric If AB = CD, then CD = AB. If m<1 = m<2, then m<2 = m<1.

4 2.6 algebraic proofs ink.notebook September 13, 2017 In geometry, a similar format is usedà two column proof or formal proof Statements and reasons organized in two columns Each step is called Statement Properties that justify each step are called Reason Geometric Proof Geometry deals with numbers as measures, so geometric proofs use properties of numbers. Here are some of the algebraic properties used in geometric proofs. Segments Angles Reflexive AB = AB m<1 = m<1 Symmetric If AB = CD, then CD = AB. If m<1 = m<2, then m<2 = m<1. Transitive If AB = CD and CD = EF, If m<1 = m<2 and m<2 = then AB = EF. m<3, then m<1 = m<3. Example 3: Write a two column proof to verify this conjecture. Given: mú1 = mú2, mú2 = mú3 Prove: mú1 = mú3 Statements Reasons 1. mú1 = mú2 Given mú2 = mú3 Given mú1 = mú3 Transitive 3. 4

State the property that justifies each statement. 1. If m 1 = m 2, then m 2 = m 1.

If AB = RS and RS = WY, then AB = WY. Transitive 4.

If m 1 + m 2 = 110 and m 2 = m 3, then m 1 + m 3 = 110. Substitution Transitive 6.

5 State the property that justifies each statement. 1. If m 1 = m 2, then m 2 = m 1. Transitive Reflexive 2. If m 1 = 90 and m 2 = m 1, then m 2 = 90. Substitution Transitive 3. If AB = RS and RS = WY, then AB = WY. Transitive 4. If AB = CD, then Reflexive Substitution Addition Multiplication 5. If m 1 + m 2 = 110 and m 2 = m 3, then m 1 + m 3 = 110. Substitution Transitive 6. RS = RS Transitive Reflexive 7. If AB = RS and TU = WY, then AB + TU = RS + WY. Substitution Addition Multiplication 8. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. Transitive Substitution 5

Practice More Practice 2.6 State the property that justifies each statement. 1. If 80 = m A, then m A = 80. 6. Complete the following proof.

6 Practice More Practice 2.6 State the property that justifies each statement. 1. If 80 = m A, then m A = Complete the following proof. Given: 8x 5 = 2x + 1 Prove: x = 1 2. If RS = TU and TU = YP, then RS = YP. 3. If 7x = 28, then x = If VR + TY = EN + TY, then VR = EN. 5. If m 1 = 30 and m 1 = m 2, then m 2 = 30. 6

7 Prove: x = 3 9. Given: 4x + 8 = x + 2 Prove: x = 2 Answers: 1. Symmetric 3. Division 5. Substitution 7. J, D, E, G, H, A, C, F, I, B 9. Given, subtraction, 3x = 2 8, 3x = 6, division, x = 2 7

8 + BOOK WORK p. 137(9 18, 43, 56 58) 8

9 9

10 Check book work answers to the odd problems in the back of the book 10

LESSON 2 5 CHAPTER 2 OBJECTIVES

LESSON 2 5 CHAPTER 2 OBJECTIVES POSTULATE a statement that describes a fundamental relationship between the basic terms of geometry. THEOREM a statement that can be proved true. PROOF a logical argument