SMSG Postulate 16. (Euclidean Parallel Postulate) Through a given external point there is at most one line parallel to a given line.
Playfair's Axiom. Through a point not on a line there is exactly one line parallel to the given line.
Playfair's Axiom is named after John Playfair (1748–1819), a Scottish physicist and mathematician, though many others had used it much earlier. Since we have shown the existence of a parallel line, it is clear that SMSG Postulate 16 (Euclidean Parallel Postulate) and Playfair's Axiom are equivalent. Further,
Theorem 2.21. In a neutral geometry,
Proof. First use
We need to show that k1 is the unique line parallel to l through P. Let k2 be another line through P such that k1 and k2 are distinct lines. Let A and B be distinct points on k2 such that A-P-B and B and Q are on the same side of k1. Let R be on l and S be on k1 such that B, R, and S are all on the same side of line PQ. Hence, since B and S are on the same side of line PQ and B and Q are on the same side of line PS, by the definition of the interior of an angle, Since line PQ is perpendicular to both l and k1, and are right angles; i.e. Since Hence, Therefore, since B and R are on the same side of line PQ and , by Euclid's Fifth Postulate, and intersect on the same side as B and R. Hence, k1 is the unique line parallel to l that contains P.
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.
Exercises 2.66. Show the Poincaré Half-plane does not satisfy
Exercise 2.67. (a) Prove five of the propositions below using the Euclidean Parallel Postulate and
Printout of the following Euclidean Propostions
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 All or None Theorem. The result is surprising since we only need existence of the parallel property for one line and one point not on the line to know that the parallel property is true everywhere. A proof of the All or None Theorem may be found in either Elementary Geometry from an Advanced Standpoint by Moise, or Geometry: A Metric Approach with Models by Millman and Parker.
Euclidean Proposition 2.2. If A and D are points on the same side of line BC and line BA is parallel to line CD, then
Euclidean Proposition 2.3. If l1, l2, l3 are three distinct lines such that l1 is parallel to l2 and l2 is parallel to l3, then l1 is parallel to l3.
Euclidean Proposition 2.4. If l1, l2, l3 are three distinct lines such that l1 intersects l2 and l2 is parallel to l3, then l1 intersects l3.
Euclidean Proposition 2.5. A line perpendicular to one of two parallel lines is perpendicular to the other.
Euclidean Proposition 2.6. If l1, l2, l3, l4 are four distinct lines such that l1 is parallel to l2, l3 is perpendicular to l1, and l4 is perpendicular to l2, then l3 is parallel to l4.
Euclidean Proposition 2.7. Every two parallel lines have a common perpendicular.
Euclidean Proposition 2.8. The perpendicular bisectors of the sides of a triangle intersect at a point.
Euclidean Proposition 2.9. There exists a circle passing through any three noncollinear points.
Euclidean Proposition 2.10. There exists a point equidistant from any three noncollinear points.
Euclidean Proposition 2.11. A line intersecting and perpendicular to one side of an acute angle intersects the other side.
Euclidean Proposition 2.12. Through any point in the interior of an angle there exists a line intersecting both sides of the angle not at the vertex.
Euclidean Proposition 2.13. If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. (The converse of Theorem 2.15.)
Euclidean Proposition 2.14. The sum of the measures of the angles of any triangle is 180.
Euclidean Proposition 2.15. There exists a triangle such that the sum of the measures of the angles of the triangle is 180.
Euclidean Proposition 2.16. The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. (Compare to the Exterior Angle Theorem.)
Euclidean Proposition 2.17. If a point C is not on segment AB but on the circle with diameter AB, then is a right angle.
Euclidean Proposition 2.18. If is a right angle, then C is on the circle with diameter AB.
Euclidean Proposition 2.19. The perpendicular bisectors of the legs of a right triangle intersect.
Euclidean Proposition 2.20. There exists an acute angle such that every line intersecting and perpendicular to one side of the angle intersects the other side.
Euclidean Proposition 2.21. There exists an acute angle such that every point in the interior of the angle is on a line intersecting both sides of the angle not at the vertex.
Euclidean Proposition 2.22. If l1, l2, l3, l4 are four distinct lines such that l1 is perpendicular to l2, l2 is perpendicular to l3, and l3 is perpendicular to l4, then l1 intersects l4.
Euclidean Proposition 2.23. There exists a rectangle.
Euclidean Proposition 2.24. There exist two lines equidistant from each other.
Euclidean Proposition 2.25. If three angles of a quadrilateral are right angles, then so is the fourth.
Euclidean Proposition 2.26. There exists a pair of similar triangles that are not congruent. (Two triangles are similar if and only if corresponding angles are congruent and the corresponding sides are proportional.)
Euclidean Proposition 2.27. The diagonals of a Saccheri quadrilateral bisect each other.
Euclidean Proposition 2.28. One of the summit angles of a Saccheri quadrilateral is a right angle.
Euclidean Proposition 2.29. Any three lines have a common transversal.
Euclidean Proposition 2.30. There do not exist three lines such that each two are on the same side of the third.
Euclidean Proposition 2.31. In , if M is the midpoint of segment AB and N is the midpoint of segment AC, then the length of segment MN is equal to half the length of segment BC.
© Copyright 2005, 2006 - Timothy Peil