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# Chapter 5: Relationships with Triangles - PowerPoint PPT Presentation

Chapter 5: Relationships with Triangles. Section 5-4: Inverses, Contrapositves , and Indirect Reasoning. Objectives. To write the negation of a statement and find the inverse and contrapositve of a conditional statement. To use indirect reasoning. Vocabulary. Negation Inverse

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### Chapter 5:Relationships with Triangles

Section 5-4:

Inverses, Contrapositves, and

Indirect Reasoning

• To write the negation of a statement and find the inverse and contrapositve of a conditional statement.

• To use indirect reasoning.

• Negation

• Inverse

• Contrapositive

• Equivalent Statements

• Indirect Reasoning

• Indirect Proof

• A negation of a statement has the opposite truth value.

• All triangles consist of 180º

• True

• Negation: All triangles do not consist of 180º

• Bethlehem is the capital of Pennsylvania.

• False

• Negation: Bethlehem is not the capital of Pennsylvania.

• We do not have school on Thanksgiving Day.

• True

• Negation: We have school on Thanksgiving Day.

• RABC is obtuse.

• Lines m and n are not perpendicular.

• The inverse of the conditional “if p then q” is “if not p then not q”

• The inverse negates both the hypothesis and conclusion.

• The contrapositive of the conditional “if p then q” is “if not q then not p”

• The hypothesis of the conditional:

• Switches the hypothesis and conclusion.

• Negates both.

Write the Inverse and the Contrapositive

• Conditional:

• 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.

• Contrapositive:

• If a figure is not a rectangle, then it is not a square.

• A conditional and its converse can have different truth values.

• Likewise, a conditional and its inverse can have different truth values.

• The contrapositive will always have the same truth value as the conditional.

• Equivalent Statements have the same truth value.

• Conditionals and contrapositives are equivalent.

• In indirect reasoning, all possibilities are considered and then all but one is proved to be false.

• The remaining possibility is true.

• You are completing a geometry problem—finding the length of a triangle side.

• You get the result: x2 = 16.

• You think through the following steps:

• You know that if x2 = 16, then x = 4 or x = -4.

• You know the length of a side is not negative.

• You conclude:___________

• A proof involving indirect reasoning is an indirect proof.

• In an indirect proof, there are often only two possibilities:

• Statement

• Negation

• Step One: State as an assumption, the opposite (negation) of what you want to prove:

• Step Three: Conclude that the assumption is false and what you want to prove must be true.

• Prove: Quadrilateral QRWZ does not have four acute angles.

• Assume: Quadrilateral QRWZ has four acute angles.

• Prove: An integer n is divisible by 5.

• Assume: An integer n is not divisible by 5.

• Identify the two statements that contradict eachother:

• VABC is acute

• VABC is scalene

• VABC is equiangular