3. Understand The Laws of Detachment and Syllogism 4. Appreciate a simple Ham Sandwich.
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1 Lesson 4 Lesson 4, page 1 of 8 Glencoe Geometry Chapter 2.2 and 2.3 If-Then Statements & Deductive Reasoning By the end of this lesson, you should be able to 1. Write a statement in if-then Form. 2. To write the converse, inverse, and contrapositive of an ifthen statement. 3. Understand The Laws of Detachment and Syllogism 4. Appreciate a simple Ham Sandwich. If-Then Statements: An if-then statement is a form of deductive reasoning that is just what the name says it is. We assume that if the if part is true, then, by the Law of Detachment, it automatically follows that the then part is always true. The previous statement is one such statement!! For example, If You are watching this show, then you will learn something about geometry. Or If D is between C and E, then CD + DE = CE (segment addition postulate)
2 These kinds of if-then statements are called conditional statements, or just conditionals. Lesson 4, page 2 of 8 An if-then statement contains two parts: The if part is called the hypothesis. The then part is called the conclusion. To symbolize an if-then statement, then let p represent the hypothesis, and q represent the conclusion. Here are some other ways to write an if-then statement: If p, then q p only if q p implies q q if p p q q is necessary for p p is sufficient for q Deductive reasoning, unlike inductive reasoning, is a valid form of proof. It is, in fact, the way in which geometric proofs are written. Here s an example: A person knows that all the men in a certain room are bakers, that all bakers get up early to bake bread in the morning, and that Jim is in that specific room. Now this statement is not written in if-then form. Sometimes rewriting a statement in this fashion can help clarify: If a person is in a certain room, then he is a baker. If a person is a baker, then he gets up early to bake that delicious bread.
3 Lesson 4, page 3 of 8 So, what can we say about Jim, if Jim is in that specific room??? Knowing these statements to be true, a person could deductively reason that Jim gets up early in the morning and has a recipe for delicious bread!! This is due to the Law of Syllogism, the Transitive property of Logical statements. If p q and q r, then p r Deductive reasoning in geometry is much like the situation described above, except it relates to geometric terms. Statement: Adjacent angles have a common side Rewriting: If two angles are adjacent, then they have a common side. Now, every statement has some relatives: The Converse of the statement is formed when the hypothesis and the conclusion are switched. q p Converse: If two angles have common side, then they are adjacent. Is this a true statement???? If we can find an example that disproves this statement, then we have found a counterexample, and we say the statement is false or not true (even though it might be true sometimes).
4 Lesson 4, page 4 of 8 To find a counterexample, we must find an example that satisfies the hypothesis, but does not satisfy the conclusion. In the diagram at right, POM!!!" and PON have common side OP, yet they are NOT adjacent. In general, the Converse is NOT TRUE! The Inverse of a statement is formed by negating both the hypothesis and the conclusion by adding the word NOT in front of them. p q or p q Statement: Vertical angles are congruent Rewriting: If two angles are vertical, then they are congruent. Converse: If two angles are congruent, then they are vertical. Inverse: If two angles are not vertical, then they are not congruent. Is the Inverse true??? To disprove it, we must find two angles are not vertical, but that ARE congruent. The same counterexample we found for the converse works here too!!
5 We can again disprove the converse with the following counterexample. Lesson 4, page 5 of 8 QOP NOP, yet they are not vertical angles. There are many examples where two angles are congruent, yet not vertical, so in general, the Inverse statement is NOT true. The contrapositive of a statement is the combination of the converse and inverse: it is the negation of the switching the hypothesis and the conclusion. q p or q p Statement: Vertical angles are congruent Rewriting: If two angles are vertical, then they are congruent. Contrapositive: If two angle are not congruent, then they are not vertical. We have already shown that the Converse and the Inverse of this statement are not necessarily true, and therefore false. But is the Contrapositive true??? To disprove the contrapositive, we must find two vertical angles that are NOT congruent Did you find them yet???? Me neither! That s because we know that ALL vertical angles are congruent, so you will never find two that are NOT!!
6 Lesson 4, page 6 of 8 Therefore, the contrapositive of a statement is ALWAYS true if the statement is true. If the original statement is TRUE, the contrapositive is TRUE. If the original statement is FALSE, the contrapositive is FALSE. They are said to be logically equivalent. The contrapositive is often more useful than the original statement when testing conditional statements. Statement: If n is an even number, then it is divisible by two. Contrapositive: If n is not divisible by two, then it is not an even number. We have now established a criteria by which all even numbers must pass in order to be bestowed the glorious title of even number. Is 17 an even number? What about those if and only if (abbreviated iff) statements? These include your postulates and definitions. Is this stated this way merely for dramatic effect? Are we just trying to be adamantly emphatic? Statement: Two angles are complementary if and only if their $ degree measures add to 90.
7 Lesson 4, page 7 of 8 If and only if, sometimes denoted by the bi-directional arrow, means that each side of the argument implies the other. In other words, the hypothesis and the conclusion are interchangeable. The hypothesis is both necessary AND sufficient for the conclusion. We can rewrite the previous iff statement two ways: 1. If two angles are complementary, then their measures add $ up to 90. $ 2. If two angles measures add up to 90, then they are complementary. This automatically means that the Converse of either statement is true. We also know the Contrapositive of both statements is true, which then implies that the Inverse of either statement is true. Isn t Logic fun??!! Say What??!! Let s obey the Laws!! Korpi The Law of Detachment and other laws of logic can be used to provide a system for reaching logical conclusions, called. A. inductive reasoning B. reasonable doubt C. detachment reasoning D. deductive reasoning ''If two numbers are even, then their sum is even'' is a true conditional, and 8 and 24 are even numbers. Use the Law of Detachment to reach a logical conclusion. A. The sum of 8 and 24 must be even. B. If the numbers 8 and 24 are odd, then their sum is 32. C. The sum of 8 and 24 must be odd. D. If the numbers 8 and 24 are even, then their sum is 32.
8 Lesson 4, page 8 of 8 ''If two angles are vertical, then they are congruent.'' is a true conditional, and 1 and 2 are vertical. Which is not true based on the Law of Detachment? A. 1 and 2 both have a measure of 45. B. Both 1 and 2 are obtuse. C. 1 and 2 are not congruent. D. 1 and 2 are congruent. Which statement follows from statements (1) and (2) by the Law of Syllogism? (1) If an object is a square, then it is a rhombus. (2) If an object is a rhombus, then it is an equilateral. A. An object is a rhombus. B. If an object is a square, then it is an equilateral. C. An object is a square. D. If an object is an equilateral, then it is a square. Which statement follows from (1) and (2) by the Law of Syllogism? (1) If two angles form a linear pair, then they are supplementary. (2) If two angles are supplementary, then the sum of their measures is 180. A. The measures of supplementary angles add up to 180. B. The sum of the measures of the angles in a linear pair is 180. C. Supplementary angles form a linear pair. D. A linear pair is formed by supplementary angles. Here s one more use (or misuse) of the Laws of Detachment and Syllogism. Can you find the error? Is there one? Nothing is better than eternal bliss. A Ham Sandwich is better than nothing. A Ham Sandwich is better than eternal bliss! (you heard it hear first!) *unless otherwise noted, all images are from ham_sandwich.jpg
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