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Chapter 5: Relationships with TrianglesPowerPoint Presentation

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
- Contrapositive
- Equivalent Statements
- Indirect Reasoning
- Indirect Proof

Negation

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

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

- RABC is obtuse.
- Lines m and n are not perpendicular.

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

- 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.

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 have the same truth value.
- Conditionals and contrapositives are equivalent.

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:

- 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

- 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:

- 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:

- 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

- Identify the two statements that contradict eachother:
- VABC is acute
- VABC is scalene
- VABC is equiangular