Chapter 2 Lesson 3 Objective: To use the Law of Detachment and the Law of Syllogism.

Deductive Reasoning: also called logical reasoning; is the process of reasoning logically from given statements to a conclusion. If the given statements are true, deductive reasoning produces a true conclusion.

Example 1: An auto mechanic knows that if a car has a dead battery, the car will not start. A mechanic begins to work on a car and finds the battery is dead. What conclusion can you make? The mechanic can conclude that the car will not start.

Example 2: A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make? The garden will be watered.

Law of Detachment: if a conditional is true and its hypothesis is true, then its conclusion is true.

Example 3: For the given true statements, what can you conclude? Given:M M Given: If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of AB. M By the Law of Detachment, you can conclude that M divides AB into two congruent segments, or AM MB.

For the given statements, what can you conclude? Given: If A is acute, then 90. is acute. Example 4:

Example 5: Does the following illustrate the Law of Detachment? Given: If it is snowing, then the temperature is less than or equal to 32°F. The temperature is 20°F. You conclude: It must be snowing. Your conclusion is true. You cannot apply the Law of Detachment and conclude that the hypothesis is true. You cannot come to any conclusion about whether it is snowing from the information given.

Example 6: Does the following argument illustrate the Law of Detachment? Given: If you make a field goal in basketball, then you score two points. Jenna scored two points in basketball. You conclude: Jenna made a field goal. NO, she could have scored two points another way.

Law of Syllogism: Law of Syllogism: allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement.

Example 7: Use the Law of Syllogism to draw a conclusion from the following true statements. If a number is prime, then it does not have repeated factors. If a number does not have repeated factors, then it is not a perfect square. You have two true conditionals where the conclusion of one is the hypothesis of the other. You can use the Law of Syllogism to draw the following conclusion: If a number is prime, then it is not a perfect square.

Example 8: If possible, state a conclusion using the Law of Syllogism. If not possible to use this law, explain why. If a number ends in 0, then it is divisible by 10. If a number is divisible by 10, then it is divisible by 5. If a number ends in 0, then it is divisible by 5.

Example 9: If possible, state a conclusion using the Law of Syllogism. If not possible to use this law, explain why. If a number ends in 6, then it is divisible by 2. If a number ends in 4, then it is divisible by 2. Not Possible; the conclusion of one statement is not the hypothesis of the other statement.

Example 10: Use the Law of Syllogism to draw a conclusion from the following true statements: If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. If a quadrilateral is a square, then it is a rectangle.

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