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Chapter 5: Relationships Within Triangles 5.4 Inverses, Contrapositives, and Indirect Reasoning

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Write the Converse: If it snows tomorrow, then we will go skiing. If two lines are parallel, then they do not intersect.

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Write two conditionals from the biconditional: A point is on the bisector of an angle if and only if it is equidistant from the sides of the angle. You will pass a geometry course if and only if you are successful with your homework.

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Negation the negation of a statement has the opposite truth value example: –“Knoxville is the capital of Tennessee” = FALSE –“Knoxville is NOT the capital of Tennessee” = TRUE

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Example 1 Write the negation of the statement: Angle ABC is obtuse. Lines m and n are not perpendicular.

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Example 1a Write the negation of the statement: The measure of angle XYZ is greater than 70. Today is not Tuesday.

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Inverses & Contrapositives inverse: –negates both the hypothesis and the conclusion contrapositive: –switches the hypothesis and the conclusion and negates both –think: “converse then inverse”

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Example 2 Write the inverse and the contrapositive: “If a figure is a square, then it is a rectangle.”

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Example 2a Write the inverse and the contrapositive: “If you don’t stand for something, then you’ll fall for anything.”

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Indirect Reasoning Think about this: Your brother tells you “Susan called a few minutes ago” You think through the following: step 1: you have two friends named Susan, Susan Brown and Susan Smith step 2: you know that Susan Brown is at band practice step 3: you conclude that Susan Smith called a few minutes ago

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Indirect Reasoning all possibilities considered all but one of the possibilities are proved false the remaining possibility must be true a proof using indirect reasoning is called an indirect proof usually in indirect proofs, a statement and its negation are the only possibilities

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Writing an Indirect Proof Step 1: –State as an assumption the opposite (negation) of what you want to prove Step 2: –Show that this assumption leads to a contradiction Step 3: –Conclude that the assumption must be false and that what you want to prove must be true

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Example 3 Write the first step of an indirect proof: Prove: Quadrilateral QRWX does not have four acute angles. First step (assume the negation):

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Example 3 Write the first step of an indirect proof: Prove: An integer n is divisible by 5. First step (assume the negation):

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Example 3a Write the first step of an indirect proof: Prove: The shoes cost no more than $20. Prove: Measure of angle A > measure of angle B.

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Example 4 Identify the two statements that contradict each other: 1. ΔABC is acute. 2.ΔABC is scalene. 3.ΔABC is equiangular.

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Example 4a Identify the two statements that contradict each other:

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Example 5 Write 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. Prove: At least one of the items costs more than $25.

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Homework

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