2.6.1 Parallel Lines without a Parallel Postulate  Acrobat Reader IconPrintout
Mathematics consists of proving the most obvious thing in the least obvious way.
 —
Exit book to another website.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.

      Two lines are parallel if and only if they do not intersect.
      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.

     
Link to video on Exterior Angle Theorem.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.//
Link to video proving the Triangle Inequality.

The Exterior Angle Theorem is used in the proof of the Triangle Inequality.

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.

Link to video on parallel lines and alternate interior angles.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  //

Illustration that the converse of the theorem is false for a neutral geometry.      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 on the left. 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   (Note the notation error in the graphic equation, it should read  
1l = {(x, y)... . The subscript is before the letter.)
      (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.
(a)  Prove that given a line and a point not on the line, there is a line parallel to the given line and passing through the point not on the line.
                        (b) 
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.

2.5.2 SAS PostulateBack to SAS PostulateNext to Saccheri Quadrilaterals2.6.2 Saccheri Quadrilaterals 

Ch. 2 Euclidean/NonEuclidean TOC  Table of Contents

  Timothy Peil  Mathematics Dept.  MSU Moorhead

© Copyright 2005, 2006 - Timothy Peil