Refresher - The Parallel Postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two dimensional geometry:
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Probably the best known equivalent of Euclid's parallel postulate is Playfair's axiom, named after the Scottish mathematician John Playfair, which states:
At most one line can be drawn through any point not on a given line parallel to the given line in a plane.

Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. This is a summary

  1. There is at most one line that can be drawn parallel to another given one by an external point. (Playfair's axiom)
  2. The sum of the angles in every triangle is 180° (triangle postulate).
  3. There exists a triangle whose angles add up to 180°.
  4. The sum of the angles is the same for every triangle.
  5. There exists a pair of similar, but not congruent, triangles.
  6. Every triangle can be circumscribed.
  7. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
  8. There exists a quadrilateral of which all angles are right angles.
  9. There exists a pair of straight lines that are at constant distance from each other.
  10. Two lines that are parallel to the same line are also parallel to each other.
  11. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).
  12. There is no upper limit to the area of a triangle. (Wallis axiom)
  13. The summit angles of the Saccheri quadrilateral are 90°.
  14. If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other

 

Euclid's fifth postulate is also known as the parallel postulate. It states:

"If a straight line intersecting two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than the two right angles."

Playfair's Axiom is often substituted for the fifth postulate, although technically it is equivalent to the fifth postulate plus the assumption that parallel lines exist, and hence is a stronger statement:

Given a line and a point not on the line, there exists exactly one line through the point that is parallel to the given line.

For a brief description of the historical context of the parallel postulate and other non-Euclidean geometries that arise when the parallel postulate is not adopted, visit: http://mathworld.wolfram.com/ParallelPostulate.html Links to an external site..

From the definitions and initial five postulates are built the whole of Euclidean geometry, such as the theorems given below and in the following pages. In the figure, line AB is parallel to line CD and these lines are cut by a transversal, line EF.

 

Corresponding Angles Theorem: Two lines cut by a transversal are parallel if and only if corresponding angles are congruent.
Note: Corresponding angles form the letter "F" in the image.