Proof of Existence of Parallel

by Russell Kennedy, University of Georgia

and Michelle Corey, Kristina Dunbar, Floyd Rinehart, UGA

Existence of Parallel:

Given a line L and a point P not on L, there exists a line M through P that is parallel to L.

Proof:

Assume we are given a point P on the line L.  Then, by the existence of perpendicular lines (from Set 1), we know that for every point A, there exists a line through A perpendicular to L.  Here, we have chosen so that A = P.  Therefore, we have a line perpendicular to L that goes through P, which is not on L.  This line is called T.

Now we can say there is an intersection where the perpendicular line T crosses the original line L, denoted B.  We are guaranteed its existence by the definition of a perpendicular line.

Since we have a point P that is not on line L but is now on the perpendicular line T, by definition of a perpendicular line through point P, we can say there exists another line M perpendicular to the first perpendicular line, T.  There is an intersection of T and M by definition, point P.

If L and M are distinct lines, a transversal of L and M is a line T which meets L and M at two distinct points (by the parallel axiom).  Next we can say that the line T is a transversal by definition.

We can now say, using the parallel axiom, that two lines L and M meet a transversal so that the sum of the interior angles of the transversal are less than 180º, then L and M intersect. 

But we know by definition that two lines do not intersect at more than one point, unless they are the same line.  To show this, consider this situation:  Let lines L and M meet at some point Q, with Q being on the same side as the points C and D (see picture above).  Let R be a point in which the distance from P to R is greater than the distance from P to Q.  In other words, pick the point R on the line M further from P. 

Now we need to consider an arbitrary line N that runs through the points B and R.  Now, we have the interior angle from T to N less than 90º, and since the definition of the parallel axiom tells us the interior angles less than 180º means we will have an intersection, we would have two intersections, namely lines Q and S.  One on this side of the transversal T at Q, and another on the opposite side of T on S.  This is a contradiction since L and M are not the same line.

Since we know that with one intersection we form the line L and point R such that we have an intersection on both sides of the transversal means there must exist a line parallel, or that has no intersection points.

Therefore, by definition of parallel, we see there exists line M through P which is parallel to L.

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