5-4 Inverses, Contrapositives and Indirect Reasoning

1. 5-4 Inverses, Contrapositives and Indirect Reasoning New Vocabulary: Negation A negation of a statement has the opposite truth value of the original statement. Example: Statement: Mitzy is a student. Negation: Mitzy is not a student. Truth value: true Truth value : false Example: Statement: The Measure of Angle ABC < 40 degrees Negation: The measure of Angle ABC > 40 degrees _

2. More vocab: Inverse: An inverse is when you take a conditional statement and negate (change the truth value of) both the hypothesis and the conclusion. Example: Conditional: If Rachelle studies for the test, then she will pass the test. Inverse: If Rachelle does not study for the test, then she will not pass the test. Notice the truth values remained the same. This is not always The case. Example: If a figure is a square, then it is a rectangle. Inverse: If a figure is not a square, then it is not a rectangle. The conditional statement is true, but the inverse is false.

3. More vocab: Contrapositive: The contrapositive of a conditional statement does two things. It switches the hypothesis and the conclusion, and negates them both. In other words, the contrapositive is the inverse of the converse. Example: Conditional: If Kevin eats twenty twinkies, then he will get an upset stomach. Contrapositive: If Kevin does not have an upset stomach, then he did not eat twenty twinkies. Contrapositives ALWAYS have the same truth value of the original conditional statement.

4. Contrapositives continued: Example: If an angle is a straight angle, then its measure is 180 degrees. Contrapositive: If an angle is not 180 degrees, then it is not a straight angle. Truth values are the same. This is always true of contrapositive statements.

5. Summing this up Symbolically: Statement Example Symbolic form Read Conditional If an angle is bisected, p -------> q If p, then q then it is divided into two congruent angles. Negation (of p) An angle is not bisected p Not p ~ ~ ~ Inverse If an angle is not bisected p ---> q If not p, then it is not divided into then not q. two congruent angles. ~ ~ Contrapositive If an angle is not divided q -----> p If not q, into two congruent angles, then not p. then it has not been bisected.

6. 5-4 Part II Contradiction: a contradiction consists of a logical incompatibility between two or more propositions. Example: 1. Triangle ABC is equiangular. 2. Triangle ABC is scalene. 3. Triangle ABC is acute. Statement 1 contradicts statement 2.

7. Indirect Reasoning A type of reasoning in which All possibilities are considered, and then all but One are proved false. Example: Your Mom says,”Adriana called a few Minutes ago.” Step 1: You have two friends named Adriana Step 2: You know one of them is at soccer practice Step 3: You conclude that the other Adriana must have Been the caller.

8. Indirect Proofs These are proofs that use indirect Reasoning. Step 1. The first step in any indirect proof is to make an Assumption that the opposite (negation) of what you are Trying to prove is correct. Step 2. Next you need to show that this assumption leads To a contradiction. Step 3. Conclude that the assumption must be false, so what You want to prove must be true.

9. Example: Prove that an obtuse triangle cannot contain a right angle. Step 1. Assume that an obtuse triangle can have a right angle Step 2. There is a contradiction here because every triangle Contains 180 degrees. Obtuse triangles have one angle greater Than 90 degrees, and if you add 90 degrees to this, you have More than 180 degrees. Step 3: Conclusion : Since my assumption was false, then Obtuse triangles cannot contain a right angle.