The Parallel Postulate

The parallel postulate, also known as Euclid's fifth postulate, states:

Given a line r and a point P not on the line, there exists exactly one line parallel to r that passes through point P.

This is considered a postulate because the uniqueness of the parallel line is not derived from other theorems or properties of the line. It is accepted as a fundamental truth.

However, the existence of a line parallel to r passing through point P can be demonstrated using the parallel lines theorem by finding a pair of congruent alternate interior angles α≅β.

Note: This postulate is known as Euclid's fifth postulate. However, its formulation is attributed to Proclus, a Byzantine mathematician and philosopher who lived in the 5th century AD.

The Proof

Consider a line r and a point P in the plane that does not lie on the line.

Choose a point P' on the line.

Then draw a line t passing through points P' and P.

Line t forms an interior angle alpha (α) with line r.

Draw an arc centered at P' with radius PP' that intersects line r at point A.

Center the compass on point P and with the same radius (PP'), draw a second arc that passes through point P'.

Connect points A and P with a segment AP.

Draw a third arc centered at P' with radius AP that intersects the second arc at point B.

Then connect points P' and B with segment BP'.

Finally, draw a line s passing through points B and P.

Two triangles, APP' and BPP', are formed between lines r and s.

According to the third criterion for triangle congruence, the triangles APP' and BPP' are congruent because they have sides of equal length.

$$ APP' \cong BPP' $$

Since they are congruent, triangles APP' and BPP' have congruent angles in corresponding positions.

Therefore, angle alpha is congruent to angle beta.

$$ \alpha \cong \beta $$

Angles alpha and beta are congruent alternate interior angles (α≅β).

Thus, by the parallel lines theorem, lines r and s are parallel.

This demonstrates the existence of line s parallel to line r passing through point P

Note: As mentioned earlier, the uniqueness of the parallel line s cannot be proven through other properties. Therefore, it is accepted as a postulate.

And so forth.