Warm-Up 1) Write each conditional statement in If-Then form.   a) All triangles have three angles. b) Every Thursday Julia goes swimming. 2) Write the converse, inverse, and contrapositive of each conditional. Determine if the converse(only) is true or false. If it is false, give a counterexample. a) If a figure is a rectangle, then it has four sides. b) If a number is divisible by 6, then it is divisible by 2.

1) Write each conditional statement in If-Then form.   All triangles have three angles. If it is a triangle, then it has three angles. b) Every Thursday Julia goes swimming. If it is Thursday, then Julia goes swimming.

2) Write the converse, inverse, and contrapositive of each conditional 2) Write the converse, inverse, and contrapositive of each conditional. Determine if the converse (only) is true or false. If it is false, give a counterexample.   If a figure is a rectangle, then it has four sides. Converse: If it has four sides, then the figure is a rectangle. False; A rhombus has four sides. Inverse: If a figure is not a rectangle, then it does not have four sides. Contrapositive: If it does not have four sides, then it is not a rectangle. b) If a number is divisible by 6, then it is divisible by 2. Converse: If a number is divisible by 2, then it is divisible by 6. False; 4 is divisible by 2 but not 6. Inverse: If a number is not divisible by 6, then it is not divisible by 2. Contrapositve: If a number is not divisible by 2, then it is not divisible by 6.

Chapter 2 Section 4 Algebra Proofs

Properties of Equality for Real Numbers Vocabulary Properties of Equality for Real Numbers Reflexive Property For every number a, a = a Symmetric Property For all numbers a and b, if a = b, then b = a Transitive Property For all numbers a, b and c, if a = b, and b = c, then a = c Addition and Subtraction Property For all numbers a, b and c, if a = b, then a + c = b + c, and a - c = b - c Multiplication and Division Property For all numbers a, b and c, if a = b, then a * c = b * c, and if c ≠ 0, a/c = b /c Substitution Property For all numbers a and b, if a = b, then a may be replaced by b in any equation or expression Distributive Property For all numbers a, b and c, a(b + c) = ab + ac

Vocabulary Cont. Property Segments Angles Reflexive PQ =PQ m<1 = m<1 Symmetric If AB = CD, then CD = AB If m<A = m<B, then m<B = m<A Transitive If GH = JK and JK = LM, the GH = LM If m<1 = m<2 and m<2 = m<3, then m<1 = m<3

Example 1) Name the property of equality that justifies each statement. a) If 3x = 120, then x = 40. Division Property of Equality    b) If 12 = AB, then AB = 12. Symmetric Property of Equality  c) If AB = BC, and BC = CD, then AB = CD. Transitive Property of Equality  d) If y = 75, and y = m<A, then m<A = 75. Substitution Property of Equality

Example 2) Name the property of equality that justifies each statement. a) If AB + BC =DE + BC, then AB = DC Subtraction Property of Equality    b) m<ABC = m<ABC Reflexive Property of Equality  c) If XY = PQ and XY = RS, then PQ = RS Substitution Property of Equality  d) If 1/3 x = 5, then x = 15 Multiplication Property of Equality e) If 2x – 9, then x = 9/2 Division Property of Equality

Example 3) Justify each step in solving (x/3) + 4 = 1 Statements Reasons (x/3) + 4 = 1 Given Subtraction Property of Equality (x/3) = -3 Multiplication Property of Equality x = -9

Example 4) Justify each step in solving (3x + 5)/2 = 7. Statements Reasons (3x + 5)/2 = 7 Given Multiplication Property of Equality 2((3x + 5)/2) = 2(7) Distributive Property of Equality 3x + 5 = 14 Subtraction Property of Equality 3x = 9 x = 3 Division Property of Equality

Given: <ABD and <DBC are complementary. Example 5) Justify the steps for the proof of the conditional If <ABD and <DBC are complementary, then <ABC is a right angle. A D C B Given: <ABD and <DBC are complementary. Prove: <ABC is a right angle. Statements Reasons <ABD and <DBC are Complementary Given m<ABD + m<DBC = 90 Definition of complementary angles m<ABD + m<DBC = m<ABC Angle Addition Postulate m<ABC = 90 Substitution Property of Equality <ABC is a right angle Definition of right angle

Example 6) Justify the steps for the proof of the conditional If PR = QS, then PQ = RS. Given: PR = QS Prove: PQ = RS Statements Reasons PR = QS Given PQ + QR = PR QR + RS = QS Segment Addition Postulate PQ + QR = QR + RS Substitution Property of Equality PQ = RS Subtraction Property of Equality