2.1.3 Using Deductive Reasoning to Verify Conjectures (continued)

In order for a conjecture to be valid by the Law of Syllogism, two conditionals where the same statement is used in the hypothesis of one and in the conclusion of the other must be true, and the conjecture must match the conditional formed by the non-matching hypothesis and conclusion. In this example, two conditionals are given to be true, and the conclusion of the first conditional is the same statement as the hypothesis of the second conditional, “ ∠ A is obtuse.” So, if the conjecture is the conditional formed by the hypothesis of the first conditional and the conclusion of the second, then the conjecture must hold by the Law of Syllogism. Notice that the conjecture’s hypothesis is the hypothesis of the first conditional and the conjecture’s conclusion is the conclusion of the second conditional. Therefore, the conjecture is valid by the Law of Syllogism. The first given conditional’s conclusion is “it is a multiple of 5.” The second given conditional’s hypothesis is “the last digit of a number is 0,” which does not match the first conditional’s conclusion. Try reversing the conditionals so that “If the last digit of a number is 0, then it is a multiple of 5” is the first conditional and “If a number is a multiple of 10, then it is a multiple of 5” is the second conditional. Now, the conclusion of the first is “it is a multiple of 5” and the hypothesis of the second is “If a number is a multiple of 10.” These statements are still not the same. Therefore, since the conclusion of neither conditional matches the hypothesis of the other conditional, this conjecture cannot be valid by the Law of Syllogism. So, even though this conjecture is true, it is not true by the Law of Syllogism.

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