Please define for me in rigorous geometric terms what it means that two lines are equidistant.Originally Posted by Nichelle14
I have to prove the following:
Prove euclidean parallel postulate iff parallel lines are equidistant from one another. My hint is this:
Use the converse to Theorem 4.1 since it can be proved assuming Neutral geometery + euclidean parallel postulate. " Parallel lines are equidistan from one another" if the length of all perpendicular segments from a point on line "dropped" to the other line are congurent[equal in length].
Observe, the picture.
As I understand, the equidistant postulate states that,
AB is perpendicular to line AD
DC is perpendicular to line BC
Also, equidistance-meaning AB=DC
Note, triangle ABD and triangle BDC
They satisfy Hypotenuse-Leg. Thus, they are congruent. Thus, <ADB=<BDC
Thus the parallel postulate is proved.
Note, the congrunce of hypotenuse-leg is independent of parallel postulate, If it were not then you cannot use it.