Reasoning in Algebra and Geometry Section 2.5

How to justify Algebra and Geometry We use properties of equality, properties of congruence, and postulates to justify each step that is taken. Proof A convincing argument that uses deductive reasoning. Two ways to write proofs Two-column proof Lists statements on the left and reasons for each statement on the right. Paragraph proof The statements and reasons are written in sentences.

Properties of Equality Addition Property Subtraction Property Multiplication Property Division Property Reflexive Property Symmetric Property Transitive Property Substitution Property If a=b, then a + c = b + c If a=b, then a – c = b – c If a=b, then a * c = b * c If a=b and c ≠ 0, then a/c = b/c a = a If a = b, then b = a If a = b and b = c, then a = c If a = b, then b can replace a in any expression.

The Distributive Property Use multiplication to distribute a to each term within the parentheses. a(b + c) = ab + ac a(b – c) = ab – ac

Properties of Congruence Reflexive Property Symmetric Property Transitive Property If , then If and , then If and , then If and , then

Example 1 What is the value of x? Justify each step. Angle AOM and angle MOC = 180 Angles form a linear pair Definition of supplementary lines (2x + 30) + x = 180 Substitution Property 3x + 30 = 180 Addition Property - 30 - 30 Subtraction Property 3x = 150 Division Property 3 3 x = 50 Substitution Property x (2x + 30) A M O C

Example 2 What is the name of the property of equality or congruence that justifies going from first to second statement? 3(x + 5) = 9 3x + 15 = 9 ¼ x = 7 x = 28

Example 3 Two-Column Proof Given: 5x + 1 = 21 Prove: x = 4 Statements Reasons . 5x + 1 = 21 Given 5x = 20 Subtraction Prop of Equality X = 4 Division Prop. of Equality

Assignment pg 117-118 #5-19 Show work