September 16

Parallel Lines and More Proofs

Euclid's Parallel Postulate | Equivalent Formulations | Names of Angles
Hypothesis and Conclusions

Euclid's Parallel Postulate

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For every line l and for every point P that does not lie on ,
there exists a unique line m through P that is parallel to .

The enormous controversy about this postulate is caused because the postulate uses the term "line," not segment. Since lines are infinite, there is no way that this postulate can be verified. As early as Proclus (410-485 AD), this postulate was under attack. Proclus's example is a hyperbola which, although it becomes closer and closer to its asymptote, it never meets it. Thus, would not a hyperbola and its asymptote be examples of parallel lines?

The problem then seems to be: what is the definition of parallel? The text states, on page 21, that "two lines that don't intersect are called parallel." Using that as a definition, then the hyperbola and its asymptote are parallel.

The proof in the text, also on page 21, shows that one line can be constructed through a point P to a line . Hilbert, a foremost mathematician of the early 20th century, rephrased Euclid's Parallel Postulate to state:

For every line and for every point P that does not lie on ,
there exists at most one line m through P that is parallel to .

This formation of the Parallel Postulate is valid for Euclidean Geometry!

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Equivalent Formulations

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The Parallel Postulate for Euclidean Geometry has several formulations, which will be discussed as the course proceeds:
1) For every line l and for every point P that does not lie on ,
there exists at most one line m through P that is parallel to .

2) The sum of the angles in a triangle is 180o.

3) Given a quadrilateral with three right angles (known as the Lambert quadrilateral), the fourth angle must be a right angle.

4) Give a quadrilateral whose base angles are right angles and whose base-adjacent sides are congruent to each other, (known as the Saccheri quadrilateral), then the summit angles must be right angles

5) Rectangles exist.

6) Given any triangle ABC and any segment , there exists a triangle DEF ( having the segment (as one of its sides) that is similar to triangle ABC.

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Names of Angles Formed by Two Parallel Lines

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< 3 and < 6 are called alternate interior angles and are congruent. (as well as < 4 and < 5)

< 1 and < 8 are called alternate exterior angles and are congruent. (as well as< 2 and < 7 )

< 1 and < 5 are called corresponding angles and are congruent. (as well as < 3 and < 7 AND < 2 and < 6 AND < 4 and < 8)

Exterior Angles on the same side of the transversal are supplementary angles, such as < 1 and < 7. (as well as < 2 and < 8)

Interior Angles on the same side of the transversal are supplementary angles, such as < 3 and < 5. (as well as < 4 and < 8)

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Hypothesis and Conclusion

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The hypothesis in a proof is that which is given to use. The conclusion is that which is to be proven. For example, the statement: "a parallelogram ABCD is a rectangle if and only if the diagonals are congruent" is actually two statements to be proven:
1) Given that parallelogram ABCD is a rectangle, prove that its diagonals are congruent.

AND

2) Given that parallelogram ABCD has congruent diagonals, prove that it is a rectangle.

Often, in developing a proof, it helps to write out both the hypothesis and the conclusion as separate statements. For example, in the above example, item #1 could be rewritten as:

Hypothesis: Parallelogram ABCD is a rectangle.

Conclusion: Its diagonals are congruent.

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