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2-3 Laws of Logic The laws of logic can be used to provide a system for reaching logical conclusions. This system is called deductive reasoning. Deductive reasoning is one of the cornerstones of the study of geometry. Law of Detachment If p → q is a true conditional statement and p is true, then q is true. p→q p q Example 1 1. If a student is enrolled at East Clarendon High, then the student has an ID number. 2. Brandon Shaw is enrolled at East Clarendon High. 3. Brandon Shaw has an ID number. Example 2 1. If an angle measures 123, then it is obtuse. 2. mC = 123. 3. C is obtuse. Determine if statement 3 follows from statements 1 and 2 by the law of detachment. 1. If you are not satisfied with a tape, then you can return it within a week for a full refund. 2. Joe is not satisfied with a tape. 3. Joe can return the tape within a week for a full refund. 1. If a rectangle has four congruent sides, then it is a square. 2. A square has diagonals that are perpendicular. 3. A rectangle has diagonals that are perpendicular. Law of Syllogism If p → q and q → r are true conditional statements, then p → r is true. p→q q→r p→r Example 1 1. If a number is a whole number, then it is an integer. 2. If a number is an integer, then it is a rational number. 3. If a number is a whole number, then it is a rational number. Example 2 1. If fossil fuels are burned, the acid rain is produced. 2. If acid rain falls, wildlife suffers. 3. If fossil fuels are burned, then wildlife suffers. Determine if statement 3 follows from statements 1 and 2 by the law of syllogism. 1. If Matt lives in Tampa, then he lives in Florida. 2. If Matt lives in Florida, then he lives in the USA. 3. If Matt lives in Tampa, then he lives in the USA. 1. A vocalist can read music. 2. Ann can read music. 3. Ann is a vocalist. Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If you plan to attend the University of Notre Dame, then you need to be in the top 10% of your class. Rosita Nathan plans to attend Notre Dame. Rosita Nathan needs to be in the top 10% of her class. If an angle has a measure less than 90, it is acute. If an angle is acute, then its supplement is obtuse. If an angle has a measure less than 90, then its supplement is obtuse. Vertical angles are congruent. <1 = <2 <1 and <2 are vertical Determine if a valid conclusion can be reached from the two true statements using the Law of Detachment or the law of Syllogism. If you want good health, then you should get 8 hours of sleep each day. Patricia Gorman wants good health. Patricia Gorman should get 8 hours of sleep each day. Detachment If AB = BC and BC = CD, then AB =CD. AB = CD No conclusion If the measure of an angle is less than 90, then it is acute. If an angle is acute, then it is not obtuse. If the measure of an angle is less than 90, then it is not obtuse. Syllogism If there are two points, then there is exactly one line that contains them. There exist two points A and B. There is exactly one line that contains points A and B. Detachment.