Question

Write Playfair’s Axiom as an equivalent version of Euclid’s fifth postulate.

Answer 1

Hint: Euclid called the obvious universal truths or assumptions that are specific to geometry as postulates. There are totally five postulates in his book ‘Elements’. His fifth postulate is about two lines and the condition at which they will coincide.

Complete answer:
Euclid’s fifth postulate is about the intersection of two lines. It states that, “If a straight line falls on two straight lines and the interior angles on the same side are less than two right angles when added together, the two straight lines would intersect on the side where the number of angles is less than two right angles if the lines are extended indefinitely.” This postulate can be understood more clearly by looking at the diagram below.

According to the fifth postulate, the lines $\overline {AB} $ and $\overline {CD} $ will intersect in the side, where the angles $\angle 1{\text{ and }}\angle 2$ are formed as we can see from the figure that the sum of these angles is less than two right angles which is $180^\circ $.
The fifth postulate also tells us that two lines would be parallel if the sum of the angle is equal to two right angles. So, axioms that talk about parallel are also equivalent to this postulate.
Euclid’s fifth postulate has more than one equivalent version. Playfair’s axiom is one such version. Playfair’s axiom says that “For every line $l$ and every point $P$ not lying on $l$, there exists a unique line $m$ going through $P$ and parallel to $l$.”
Another variant of the preceding postulate is “Two intersecting distinct lines cannot be parallel to the same line.”

Note:
The fifth postulate was not needed to prove Euclid's first 28 theorems. Many mathematicians, like Euclid, believed the fifth postulate is a theorem that could not be proven. No one has been able to prove the fifth postulate despite numerous attempts.