Each of the following is an equivalent Euclidean postulate.
Equivalent Euclidean Postulates:
- (Playfair) Given a line and a point not on that line, there exists exactly one line through that point parallel to the given line.
- (Equidistance) Lines that are parallel are everywhere equidistant.
- (Euclid) Given two lines and a transversal of those lines, if the sum of the angles on one side of the transversal is less than two right angles then the lines meet on that side.
These theorems do not require a parallel postulate.
If the alternate interior angles formed by a transversal of two lines are equal, then the lines are parallel.
The (shortest) distance between a point and a line is from the point to the foot of the perpendicular.
Given two lines and a transversal of those lines, if the sum of the angles on one side of the transversal is equal to two right angles then the lines are parallel.
The following theorem produces an easier to use version of Euclid's postulate.
The alternate interior angle converse theorem states Given parallel lines and a transversal of those lines, the alternate interior angles formed by the transversal are congruent.''