1. 3.1 Indirect Proof and Parallel Postulate
CHAPTER
3.1 Indirect Proof and Parallel Postulate
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Definition
In this section, we learn a new type of reasoning called indirect reasoning. In indirect reasoning, all possibilities are considered and then all but one are proved false. The remaining possibility must be true. A proof involving indirect reasoning is an indirect proof.
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3. How to Write an Indirect Proof
Step 1. Identify the statement that we are trying to prove true. Step 2. Assume that the opposite of the statement in Step 1 is true. Step 3. Logically try to prove that the statement in Step 2 is true, but a contradiction should occur. Step 4. If we arrive at a contradiction, then the statement in Step 2 is false. Thus, the original statement that we wanted to prove (in Step 1) must be true.
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4. Writing the Second Step of an Indirect Proof
Suppose you want to write an indirect proof of each statement. As the second step of the proof, what would you assume? a. An integer n is divisible by 5. b. You do not have soccer practice today. Solution a. The opposite of “is divisible by” is “is not divisible by.” Assume temporarily that n is not divisible by 5.
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5. Writing the Second Step of an Indirect Proof
Suppose you want to write an indirect proof of each statement. As the second step of the proof, what would you assume? a. An integer n is divisible by 5. b. You do not have soccer practice today. Solution b. The opposite of “do not have” is “do have.” Assume temporarily that you do have soccer practice today.
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6. Identifying Contradictions
Which two statements contradict each other? I. II. III. Solution Statements I and II: Segments can be parallel and congruent. Thus, statements I and II do not contradict each other. Statements II and III: Segments can be congruent and perpendicular. Thus, statements II and III do not contradict each other.
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7. Identifying Contradictions
Which two statements contradict each other? I. II. III. Solution Statements I and III: Parallel segments do not intersect, so they cannot be perpendicular. Thus, statements I and III DO contradict each other.
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8. Significant Theorems Proved Indirectly
Square root of 2 is an irrational number Fermat’s Last Theorem No Least Positive Rational Number When you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth.” - Sir Arthur Conan Doyle,
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9. Postulate 3.2-1 Parallel Postulate
Through a point not on a line, there is one and only one line parallel to the given line.
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10. Postulate 3.2-2 Perpendicular Postulate
Through a point not on a line, there is one and only one line perpendicular to the given line.
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11. Theorem 3.4-1 Perpendicular Transversal Theorem
Theorem In a plane, let two parallel lines be cut by a transversal. If the transversal is perpendicular to one of the parallel lines, then it is perpendicular to the other parallel line. If… Then…
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12. Theorem 3.4-2 Two Lines Parallel to a Third Line
Theorem If two lines are parallel to the same line, then all three lines are parallel to each other. If… Then…
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13. Proving Perpendicular Transversal Theorem
Prove Theorem Given: In a plane, l || m and Prove:
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14. Proving Perpendicular Transversal Theorem
Statements
Reasons
1. Given
2. Corresponding Angles Converse
3. definition of right angles
4. definition of perpendicular lines
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