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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.
Proof: Consider any triangle, say ABC A B C. At A on AB, and on the opposite side, copy ∠ABC ∠ A B C, say ∠DAB ∠ D A B, and at A on AC, and on the opposite side, copy ∠ACB ∠ A C B to obtain ∠EAC ∠ E A C. Why is line DA = AE D A = A E so that α + β + γ = 180∘ α + β + γ = 180 ∘?
The parallel postulate tells us that if the two labeled angles don't add up to exactly 180° in measure, the two lines u and v aren't parallel. One of the labeled angles is 100° and the other is a right angle (or 90°).
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:
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.) Thus, in the presence of the axioms of neutral geometry, the Euclidean Parallel Postulate and Euclid’s Postulate V are equivalent, meaning that each one implies the other.
For over two millenia mathematicians tried to prove Euclid's parallel postulate from the other four of his postulates. This was known early on to be a useless effort, but it was not known until the 19th century why they were right.
Euclid's 5th postulate, also known as the parallel postulate, is an important part of geometry because it helps to define what it means for two lines to be parallel. In Euclidean geometry, two lines are parallel if they never intersect, no matter how far they are extended.
