Chapter 5 relationships with triangles
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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

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Chapter 5 relationships with triangles

Chapter 5:Relationships with Triangles

Section 5-4:

Inverses, Contrapositves, and

Indirect Reasoning


Objectives

Objectives

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

  • To use indirect reasoning.


Vocabulary

Vocabulary

  • Negation

  • Inverse

  • Contrapositive

  • Equivalent Statements

  • Indirect Reasoning

  • Indirect Proof


Negation

Negation

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


Examples

Examples:

  • 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.


Write the negation

Write the Negation

  • RABC is obtuse.

  • Lines m and n are not perpendicular.


Inverse

Inverse

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

  • The inverse negates both the hypothesis and conclusion.


Contrapositive

Contrapositive

  • 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

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.


Recall

Recall:

  • 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

Equivalent Statements

  • Equivalent Statements have the same truth value.

  • Conditionals and contrapositives are equivalent.


Indirect reasoning

Indirect Reasoning

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

  • The remaining possibility is true.


Example

Example:

  • 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:___________


Indirect proof

Indirect Proof

  • A proof involving indirect reasoning is an indirect proof.

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

    • Statement

    • Negation


Writing an indirect proof

Writing an Indirect Proof:

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

  • Step Two: Show the assumption leads to a contradiction.

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


The first step of an indirect proof

The first step of an indirect proof:

  • 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.


Identifying contradictions

Identifying Contradictions

  • Identify the two statements that contradict eachother:

    • VABC is acute

    • VABC is scalene

    • VABC is equiangular


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