Warm-Up: EOC Prep What is RD if RL=54 cm? A) 81 B) 108 C) 162 What is x if WL=15x and D) 216 LJ=5x+3? A) 0.3 B) 0.4 C) 0.6 D) 1.2 Inverses, Contrapositives, and Indirect Reasoning 5.4 Today’s Goals By the end of class today, YOU should be able to… 1. Write the negation of a statement and the inverse and contrapositive of a conditional statement. 2. Use indirect reasoning to solve a problem. Negations A negation of a statement has the opposite meaning of the original statement. Ex.1: Negations Write the negation of the following statement… Statement: ABC is obtuse. Ex.1: Solution Negation: ABC is not obtuse. You Try… Write the negation of the following statement… Statement: Lines m and n are not Inverses and contrapositives A conditional statement is a statement that tells if one thing happens, another will follow It is written as p q The inverse of a conditional statement negates both the hypothesis and the conclusion. The contrapositive of a conditional statement switches both the hypothesis and conclusion and negates both. Ex.2: Write the inverse and contrapositive of the following statement: Statement: If a polygon has six sides, then it is a hexagon. Ex.2: Solution Conditional: If a polygon has six sides, then it is a hexagon. Negate both. Inverse: If a polygon does not have six sides, then it is not a hexagon. Conditional: If a polygon has six sides, then it is a hexagon. Switch and negate both. Contrapositive: If a polygon is not a hexagon, then it does not have six sides. You Try… Write the inverse and contrapositive of the following statement: If a polygon is a triangle, then the sum of the measures of its angles is 180. A summary of negation, inverse, and contrapositive statements Statement Example Symbolic form You read it Conditional If an angle is a straight angle, then its measure is 180. pq If p, then q Negation (of p) An angle is not a straight angle ~p Not p Inverse If an angle is not a straight angle, then its measure is 180 ~p ~q If not p, then not q Contrapositive If an angle’s measure is not 180, then it is not a straight angle ~q ~p If not q, then not p Indirect reasoning Indirect reasoning is when all possibilities are considered and then all but one are proved false. Then the remaining possibility must be true. A proof involving indirect reasoning is an indirect proof. An example of indirect reasoning… Suppose your brother tells you, "Susan called a few minutes ago." You think through these three steps. 1. You have two friends named Susan. 2. You know that one of them is at band practice. 3. You conclude that the other Susan must have been the caller. Writing an indirect proof 1. 2. 3. State as an assumption the opposite (negation) of what you want to prove. Show that this assumption leads to a contradiction. Conclude that the assumption must be false and that what you want to prove must be true. Ex.3: Indirect proofs Write the first step of an indirect proof: Quadrilateral QRWX does not have four acute angles. Ex.3: Solution Quadrilateral QRWX does not have four acute angles. Assume that quadrilateral QRWX has four acute angles. You Try… Write the first step of an indirect proof: An integer n is divisible by 5. Ex.4: Indirect proofs Read the conditional statement. Think about what is given and what you are to prove. Then give the steps of an indirect proof. If Jaeleen spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25. Ex.4: Solution Given: The cost of two items is more than $50. This means that each item costs $25 or less. Prove: At least one of the items costs more than $25. Step 1: Assume as true the opposite of what you want to prove. That is, assume that neither item costs more than $25. Step 2: This, in turn, means that the two items together cost $50 or less. This contradicts the given information that the amount spent is more than $50. Step 3: Conclude that the assumption is false. One item must cost more than $25. Homework Page 267 #s 8, 13, 15, 16 Page 268 #s 21, 24-27, 31 The assignment can also be found online at: • http://www.pearsonsuccessnet.com/snpap p/iText/products/0-13-037878-X/Ch05/0501/PH_Geom_ch05-01_Ex.pdf

1/--страниц