** 2.7.1 Euclidean Parallel Postulate
***Printout*

*This ought even to be struck out of the Postulates
altogether; for it is a theorem involving many difficulties.
—*

*
**Euclid**'s
Fifth Postulate** .* That, if a straight line falling on two straight lines
make the interior angles on the same side less than two right angles, the two
straight lines, if produced indefinitely, meet on that side on which are the
angles less than the two right angles.

Playfair's Axiom is named after

We need to show that

Next, use the Euclidean Parallel Postulate to prove

We need to show that ray *BA*
intersects ray *CD*. By the
Angle
Construction Postulate, there is a ray *BE*
with *E* and *A* on the same side of line *BC*
such that

Let *F* be a point
such that *E-B-F*, then and are a linear pair. Hence and are supplementary. Hence,

By (2) and (3), .
Hence Since *D*
and *F* are on opposite sides of line *BC*, and are alternate interior angles. Hence by
Theorem 2.15, line *EB* is parallel to
line *DC.* By (1) and (2), Hence line *AB*
and line *EB* are distinct lines
through *B*. Thus, by the Euclidean
Parallel Postulate, line *AB* is not
parallel to line *DC.*

By Theorem 2.7, since ,
we have Thus, since *A* and *C* are on the same
side of line *EB*, and *C*
are on the same side of line *EB*.
Since line *EB* and line *DC* are parallel, line *DC* is on one side of line *EB*. Hence ray *BA *intersects line *CD*.

Since *A* and *D* are on the same
side of line *BC*, and are on the same side of line *BC*. Hence, ray *BA* intersects ray *CD*.//

There are many statements that are equivalent to the Euclidean Parallel Postulate, which could be used as the axiom. We list several of them below after the exercises. How many of them can you show are equivalent? The exercises ask you to prove one direction on a few of the statements and to find a counter-example in the Poincaré Half-plane.

** Exercises
2.65. ** Show the Poincaré Half-plane
does not satisfy the Euclidean Parallel Postulate. (a) Use dynamic geometry
software to construct an example. (b) Find
an analytic example.

** Euclidean Proposition 2.1.** There exists a line and a point not
on the line such that there is a unique line through the point that is parallel
to the line.

Note how this proposition differs from Playfair's Axiom. This proposition only says that at least one such point and line exist; whereas, Playfair's Axiom says that it is true for every line and point not on the the line. The surprising result that in a neutral geometry this proposition implies Playfair's Axiom is called the

Euclidean Proposition 2.2.

Euclidean Proposition 2.3.

2.6.2 Saccheri Quadrilateral2.7.2 Hyperbolic Parallel Postulate |

© Copyright 2005, 2006 - Timothy Peil |