2.6.1 Parallel Lines without a Parallel Postulate
Mathematics consists of proving the most obvious thing in the least obvious way.
 —George Polyá (1887–1985)

 

      Before adding a parallel postulate to our study, we consider several questions about parallel lines. Do parallel lines exist? How can we prove two lines are parallel? What do we know about parallel lines without a parallel postulate?

Definitions.
      Given , if A-C-D, then  is an exterior angle of  Also,  and  are called remote interior angles.
      Given line AB, line DE, and line BE such that A-B-C, D-E-F, and G-B-E-H where A and D on the same side of line BE, then line BE is called a transversal. Angles  and  (also  and  ) are called alternate interior angles.

      The next theorem will be useful in proving two lines are parallel. From your high school geometry course, you may remember the theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. That result follows from a Euclidean parallel postulate; however, the following theorem holds in any neutral geometry.

Theorem 2.11. (Exterior Angle Theorem) Any exterior angle of a triangle is greater in measure than either of its remote interior angles.

Outline of the proof. Let  be given. Let D be a point such that A-C-D, i.e.  is an exterior angle of .
1. Let M be the midpoint of segment BC.
2. B-M-C and
3. There is a point E on ray AM such that A-M-E and ME = MA.
4.
5.  and  are vertical angles.
6.
7.
8.
9.
10. E and D are on the same side of line BC.
11. B, M, E are on the same side of line CD.
12.
13.
14.
15. The proof of the case for the other remote interior angle is similar.//

Theorem 2.12. Given a line and a point not on the line, there exists a unique line perpendicular to the given line through the given point.

Theorem 2.13. Two lines perpendicular to the same line are parallel.

Theorem 2.14. There exist at least two lines that are parallel to each other.

Theorem 2.15. If there is a transversal to two distinct lines with alternate interior angles congruent, then the two lines are parallel.

Outline of the proof.  Assume line AB, line DE, and line BE are distinct with A-B-C, D-E-F, and G-B-E-H. Further, assume A and D are on the same side of line BE. We will prove the contrapositive; therefore, assume line AB and line DE are not parallel.
1. There exists a point P at the intersection of line AB and line DE.
2. Without loss of generality, assume P is on the same side of line BE as is A.
3. Consider
4.  and  are exterior angles of
5.  and , i.e.  and  
6.  and  //

      In general for a neutral geometry, the converse of the above theorem is not valid. The converse is valid in a Euclidean geometry, which is discussed after the Euclidean Parallel Postulate. For an example where the converse is false, consider the Poincaré Half-plane and the illustration. Note that line AB is parallel to line DE, but the alternate interior angles angle ABE and angle BEF are not congruent. (Angle ABE is a right angle and angle BEF is an acute angle.)

Exercise 2.55. For the Poincaré Half-plane, find all lines parallel to the given line through the given point.
      (a)  (2, 1) and
      (b)  (2, 1) and

Exercise 2.56. Justify each step and fill in any gaps in the proof of the Exterior Angle Theorem.

Exercise 2.57. Prove Theorem 2.12. (Hint. Use SAS Postulate to prove the existence of a perpendicular line and Exterior Angle Theorem to prove uniqueness.)

Exercise 2.58. Prove Theorem 2.13.

Exercise 2.59. Prove Theorem 2.14.

Exercise 2.60. Justify each step and fill in any gaps in the proof of the Theorem 2.15.

Exercise 2.61. State and prove the AAS Theorem for congruent triangles.