Question

Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

Answer 1

Hint: Here, we will first use that the Euclid’s fifth postulate is if a straight line falling on two straight lines makes the 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 and then prove it by using the graph and angles.

Complete step-by-step answer:
We know that the Euclid’s fifth postulate is if a straight line falling on two straight lines makes the 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.
It says that if $$\mathrm{\angle }1+\mathrm{\angle }2<{180}^{\circ }$$, the line $$a$$ and $$b$$ meet on the right side of line $$c$$.

Now since we know that $$a$$ and $$b$$ are parallel, then $$\mathrm{\angle }1+\mathrm{\angle }2={180}^{\circ }$$, we have

Therefore, by Euclid’s fifth postulate, line $$a$$ and $$b$$ will not meet on the right side of $$c$$ as sum is not less than $${180}^{\circ }$$ as sum is not less than 180 degrees.
Similarly, we have $$\mathrm{\angle }3+\mathrm{\angle }4={180}^{\circ }$$.
Hence, by Euclid’s fifth postulate, line $$a$$ and $$b$$ will not meet on the left side of $$c$$ as sum is not less than $${180}^{\circ }$$ as sum is not less than 180 degrees.
Thus, Euclid’s postulate implies the existence of parallel lines.

Note: While solving these types of questions, students should know that a statement is an axiom, which is taken to be true without proof and postulates are the basic structure from which lemmas and theorems are derived. We need to know about the Euclid’s postulates before finding the solution of the problem.