Warm-Up:EOC Prep

What is RD if RL=54 cm?

A) 81B) 108C) 162D) 216

What is x if WL=15x and LJ=5x+3?A) 0.3B) 0.4C) 0.6D) 1.2

Inverses, Contrapositives, and Indirect Reasoning5.4

1. Write the negation of a statement and the inverse and contrapositive of a conditional statement.

2. Use indirect reasoning to solve a problem.

Today’s GoalsBy the end of class today, YOU should be able to…

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 perpendicular.

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.

Inverse: If a polygon does not have six sides, then it is not a hexagon.

Negate both.

Conditional: If a polygon has six sides, then it is a hexagon.

Contrapositive: If a polygon is not a hexagon, then it does not have six sides.

Switch and negate both.

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.

p q 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. State as an assumption the opposite (negation) of what you want to prove.

2. Show that this assumption leads to a contradiction.

3. 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.

Quadrilateral QRWX does not have four acute angles.

Assume that quadrilateral QRWX has four acute angles.

Ex.3: Solution

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.

Given: The cost of two items is more than $50. This means that each item costs $25 or less.

Ex.4: Solution

Step 1: Assume as true the opposite of what you want to prove. That is, assume that neither item costs more than $25.

Prove: At least one of the items 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/s

npapp/iText/products/0-13-037878-X/Ch05/05-01/PH_Geom_ch05-01_Ex.pdf