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Parallel Lines: Uniqueness, Angle-sums & Playfair’s Postulate Euclid finally invokes the parallel postulate to prove the converse of I.27, showing that the congruent alternate angle approach is the only way to have parallel lines. Theorem 2.6 (I.29). If a line falls on two parallel lines, then the alternate angles are congruent. Proof.
In geometry, the parallel postulate, also called Euclid 's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
Figure 3: Euclid’s fifth postulate. In fact, the converse is also true: If in addition to the six postulates of Neutral Geometry, we assume Euclid’s Postulate V instead of the Euclidean Parallel Postulate, then the Euclidean Parallel Postulate can be proved as a theorem. (See Exercise H2.6.)
Parallel postulate: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
Theorem: Parallel lines intersecting a circle determine congruent arcs. Proof: This is immediate from the Inscribed Angle Theorem. Note: There is a problem in that we have never discussed what congruent arcs even means; everything has been in terms of breaking a figure into congruent triangles which is impossible for arcs of circles.
To show that Euclid's and Playfair's versions said exactly the same thing, we were obliged to prove what looks like a very complicated theorem of the form \begin{eqnarray*} (H_1 \implies C_1) & \implies & ( H_2 \implies C_2) \\ \end{eqnarray*} where $H_i$ is the hypothesis, and $C_i$ is the conclusion of Euclid's ($i=1$) and Playfair ($i=2$).
Throughout the history of mathematics, attempts were made to prove this postulate or state a related postulate that would make it possible to prove Euclid’s parallel postulate. Other postulates have been pro-posed that appear to be simpler and which could pro-vide the basis for a proof of the parallel postulate.
