Mathematics

Formulate the Euclid's $${5^{th}}$$ postulate in a simpler manner.


SOLUTION
Postulate 5 : If a straight line falling on two straight lines makes that interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Here two right angles means $$180^{\circ}$$.

Ref. image I

It says if $$\angle 1 +\angle 2 = 180^{\circ}$$.

There lines AB & CD meets on left side of line PQ. 

If $$ \angle 1 +\angle 2 = 180^{\circ}$$

Then lines AB & CD do not intersect and intersecting lines PQ is perpendicular to AB & CD.          Ref. image II

We rewrite postulate 5 as

"straight lines intersecting

(i) There is a line through P which is parallel to (i) and (ii) there is only on such line.

(ii) Two distinct intersecting lines cannot be parallel to the same line.
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