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Q9.

Expert-verifiedFound in: Page 216

Book edition
Student Edition

Author(s)
Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages
227 pages

ISBN
9780395977279

Write an indirect proof in paragraph form.

Given: Transversal *t* cuts lines *a* and *b*; $m\angle 1\ne m<2$

Prove: $a\overline{)\left|\right|}b$

An indirect proof in paragraph form is-

Proof: Assume temporarily that *a* is parallel to *b*, that is., $a\parallel b$

If the two lines are parallel, then $m\angle 1$ should be equal to $m\angle 2$ as alternate exterior angles are equal, however, it is given that role="math" localid="1638446966534" $m\angle 1\ne m\angle 2$. Therefore, the assumption *a* is parallel to *b*, that is., $a\parallel b$ is wrong and therefore, $a\overline{)\left|\right|}b$.

Hence, *a* is not parallel to *b*.

An indirect proof is a proof wherein you begin by assuming temporarily that the desired conclusion is not true which then by reasoning logically reaches to a certain contradiction or some other known fact.

1. Assume temporarily that the conclusion is not true.

2. Reason logically until you reach a contradiction.

3. Point out that the assumption was wrong and the conclusion must then be true.

Consider the following: Transversal *t* cuts lines *a* and *b*; $m\angle 1\ne m<2$.

In order to write an indirect proof to prove that $a\overline{)\left|\right|}b$ assume temporarily that the conclusion above is untrue.

Proof: Assume temporarily that *a* is parallel to *b*, that is., $a\parallel b$.

If the two lines are parallel, then $m\angle 1$ should be equal to $m\angle 2$ as alternate exterior angles are equal, however, it is given that $m\angle 1\ne m\angle 2$. Therefore, the assumption *a* is parallel to *b*, that is., $a\parallel b$ is wrong and therefore, $a\overline{)\left|\right|}b$,

Hence, *a* is not parallel to *b*.

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