Equivalent Versions of Euclid's Fifth Postulate The Fifth Postulate of Euclid posits the following: > For any two distinct points A and B, there exists exac...
Equivalent Versions of Euclid's Fifth Postulate
The Fifth Postulate of Euclid posits the following:
For any two distinct points A and B, there exists exactly one line segment segment AB.
There are several equivalent versions of this postulate that can be expressed in different ways.
One Version:
For any two distinct points A and B, the distance from point A to point B is equal to the distance from point A to point C, if point C lies on the line segment AB.
Another Version:
If line segment AB intersects line segment CD at point D, then AD = CD.
A Third Version:
If we have two lines segment segments AB and CD that intersect at point A, then the angle formed at A is equal to the angle formed at C.
Equivalence of Postulates:
The Fifth Postulate is equivalent to the following two statements:
For any two distinct points A and B, there exists exactly one line segment segment AB.
For any two distinct points A and B, the distance from point A to point B is equal to the distance from point A to point C if point C lies on the line segment AB