analog spear
Knowing the postulate is equivalent to properties of transversal angles, therefore we may prove the postulate by proving any of them.
We'll redefine "parallel" as "two lines of each other's translates in opposite directions and same distance."
We'll now prove that these two lines are never intersecting. Assuming these two lines intersect at some point, then this point must be on both lines, and therefore the translate of some points from either lines. But assuming the two lines are not translating in the directing they are extending in, this point cannot exist due to definition of translate.
After proving that the two lines never intersect, and equivalent to the traditional parallel, we may now prove the properties of transversal angles. Construct a transversal line through a parallel line under our definition translated in any directions and any distance. We may translate one of the lines in one of its extending directions until the direction of the translate is the same as the direction the transversal line is extending in. And due to definition of straight lines, this new translate coincides the original position, proving the line unchanged. We'll now translate the entirety of one of the lines and the transversal line in the direction of the transversal line's extending direction towards the other line, until this line coincides with the other line, which is possible due to it being the other's translate. The translated transversal line would also coincide itself due to translating in its own extending direction. This proves the two shape of one line and the transversal line and the other line and the transversal line are congruent due to its definition, and therefore proving the corresponding angles, alternate angles are equal, and the consecutive interior angles add up to two right angles due to being one side of a line.
fast copper