Reasoning in Algebra & Deductive Reasoning (Review) Chapter 2 Section 5.

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Presentation transcript:

Reasoning in Algebra & Deductive Reasoning (Review) Chapter 2 Section 5

Review Conditional: If three points are collinear, then they lie on the same line. Converse: If three points lie on the same line, then they are collinear. Biconditional: Three points are collinear if and only if they lie on the same line.

Deductive Reasoning the process of reasoning logically from accepted facts, definitions and properties to a conclusion. Law of Detachment If a conditional statement is true and the hypothesis is true, then the conclusion is true. If a car has a dead battery, then the car will not start. Jim’s car has a dead battery. Example Conclusion: Jim’s car will not start.

Law of Syllogism Stating a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement. If p→q and q→r, then p→r. If you spend time with friends, then you enjoy yourself. If you enjoy yourself, then your time is well spent. Example Conclusion: If you spend time with friends, then your time is well spent.

Practice Use the Law of Detachment to draw a conclusion. Use the Law of Syllogism to draw a conclusion. If two lines are parallel, then they do not intersect. Line f is parallel to line m. If two planes intersect, then they intersect in a line. If two planes are not parallel, then they intersect. Lines f & m do not intersect. If two planes are not parallel, then they intersect in a line.

Properties of Equality Addition Property If a = b, then a + c = b + c Ex. x – 2 = 4 x – = x = 6 Given Addition Prop. Simplify

Properties of Equality Subtraction Property If a = b, then a + c = b + c Ex. x – 2 = 4 x – = x = 6 Given Addition Prop. Simplify

Multiplication Property If a = b, then a c = b c Ex. x / 2 = 4 2 x / 2 = 4 2 x = 8 Given Multiplication Prop. Simplify Properties of Equality

Division Property If a = b, then a / c = b / c Ex. 2x = 4 2x / 2 = 4 / 2 x = 2 Given Division Prop. Simplify Properties of Equality

Substitution Property If a = b, then b can replace a in any expression Ex.Let x = 2 and y = 3x + 4 y = 3 (2) + 4 y = = 10 Given Substitution Prop. Simplify y = 3x + 4 Properties of Equality

Distributive Property a(b + c) = ab + ac Ex. 2x + 8 Given Distributive Prop. 2(x + 4) Properties of Equality

Reflexive Property a = a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c Properties of Equality

Example 1 Solve the equation. Justify each step. Given Subtraction Prop. Of Equality Division Prop. Of Equality

Example 2:Solve for x and justify each step. Given: A O C B Angle Addition Prop Substitution Prop Simplify Subtraction Prop Division Prop

Example 3:Solve for x and justify each step. D E F G Angle Addition Prop Substitution Prop Simplify Subtraction Prop Division Prop

Example 4:Solve for x and justify each step. Given: A B C Segment Add. Prop. Substitution Prop Distributive Prop Simplify Subtraction Prop Division Prop

Assignment#23 Pg. 107 #2-34 even

Review Addition Property If a = b, then a + c = b + c Multiplication Property If a = b, then a c = b c Division Property If a = b, then a/c = b/c Substitution Property If a = b, then b can replace a in any expression Distributive Property a(b + c) = ab + ac Reflexive Property Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c a = a