Laws of Logic Law of Detachment If p  q is a true conditional statement AND p is true, then you can conclude q is true Example If you are a sophomore you take drivers ed. Steph is a sophomore. Therefore you can conclude she takes drivers ed. Law of Syllogism If p  q is true AND q  r is true then you can conclude p  r Example If a bird is fastest bird on land then it is the largest of all birds. If a bird is the largest of all birds then it is an ostrich. You can conclude that if a bird is the fastest bird on land then it is an ostrich.

Practice Problems 1 Decide whether inductive or deductive reasoning was used to reach the conclusion. Explain your reasoning. 1) For the past three Wednesdays the cafeteria has served macaroni and cheese for lunch. Dana concludes that the cafeteria will serve macaroni and cheese for lunch this Wednesday. 2) If you life in Nevada and are between the ages of 16 and 18, then you must take driver’s education to get your license. Marcus lives in Nevada, is 16 years old. To get his driver’s license Marcus took driver’s ed.

Practice Problems 2 Use the Law of Syllogism or the Law of Detachment to draw a conclusion if possible. If not possible, write “Not Valid” and explain why. 1) If the sum of the measures of two angles is 90 o, then the two angles are complementary. m<A + m<B = 90 o 2) If the sun is shining then it is a beautiful day. If it is a beautiful day, then we will have a picnic. 3) If two adjacent angles form a right angle, then the two angles are complementary. <A and <C are complementary. 4) If Ginger goes to the movies, the Marta will go to the movies. If Yumi goes to the movies, then Ginger will go to the movies.

Practice Problems 3 Use the Law of Syllogism or the Law of Detachment to draw a conclusion if possible. If not possible, write “Not Valid” and explain why. 1) If the stereo is on, then the volume is loud. If the neighbors complain about the stereo, then the volume is loud. 2) If <A and <C are acute angles, then any angle whose measure is between the measures of <A and <C is also acute. m<A < m<B < m<C. 3) If a creatures is a fly, then it has six legs. If a creature has six legs, then it is an insect.

Practice Problems 4 Use the true statements to determine whether the conclusion is true or false. Explain your answer. Be sure to reference a law of logic. –If John drives into the fence, then John’s father will be angry –If John backs the car out, then John will drive into the fence –John backs the car out 1) Does John drive into the fence? 2) Is John’s father angry?

Algebraic Properties of Equality Addition Property If a = b, then a + c = b + c Subtraction Property If a = b, then a – c = b – c Multiplication Property If a = b, then ca = cb Division Property If a = b, then a ÷ c = b ÷ c where c ≠ 0

Algebraic Properties of Equality Reflexive Property For any real number a, a = a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c Substitution Property If a = b, then a can be substituted for b in any equation or expression.

Geometric Properties of Equality Segment LengthAngle Measure Reflexive For any segment AB, AB = AB For an angle A, m<A = m<A Symmetric If AB = CD, then CD = AB If m<A = m<B, then m<B = m<A Transitive If AB = CD and CD = EF then AB = EF If m<A = m<B and m<B = m<C then m<A = m<C