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The Euclidean Parallel Axiom is as stated below:

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.

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14y ago
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6mo ago

The Euclidean Parallel Axiom states that through a point not on a given line, there exists exactly one line parallel to the given line. This axiom is one of the five postulates in Euclidean geometry that forms the foundation for the study of parallel lines and geometry.

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Q: What is the Euclidean Parallel Axiom?
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What are the basic constructions required by Euclid's postulates?

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Related questions

Why is the parallel axiom in Euclid's geometry false in non-Euclidian geometry?

Euclid's parallel axiom is false in non-Euclidean geometry because non-Euclidean geometry occurs within a different theory of space. There may be one absolute occurrence in non-Euclidean space where Euclid's parallel axiom is valid. Possibly as some form of infinity.


Which euclidean axiom has been considered controversial?

the 5th one


What is two point form of a line?

An axiom of Euclidean geometry.


Why hilbert axiom of parallelism assert the existence of only at most one parallel line'?

There is a subtle distinction between Euclidean, Hilbert and Non-Euclidean planes. Euclidean planes are those that satisfy the 5 axioms, while Non-Euclidean planes do not satisfy the fifth postulate. This means that in Non-Euclidean planes, given a line and a point not on that line, then there are two (or more) lines that contain that point and are parallel to the original line. There are geometries where there must be exactly one line through that point and parallel to the original line and then there are also geometries where no such line contains that point and is parallel to the original line.Basically, the fifth postulate can be satisfied by multiple geometries.


Do parrallel lines meet?

Not in Euclidean Geometry. Euclid's 5th axiom is that parallel lines never meet. However, unlike the first 4 axiom, it is impossible to prove the 5th axiom; depending upon the situation, you can either assume that parallel lines meet or don't; when they do meet, there are some very interesting consequences (for example, the possibility of a hyperbolic space). To my knowledge, if they meet, they are intersecting/perpendicular lines.


Are two lines that are parallel to the same line parallel to each other?

Yes they are. It is delineated in something called the parallel postulate, and the axiom is also called Euclid's fifth postulate. This is boilerplate Euclidean geometry, and a link can be found below if you'd like to review the particulars.


What is the difference between Euclidean Geometry and non-Euclidean Geometry?

In Euclidean geometry parallel lines are always the same distance apart. In non-Euclidean geometry parallel lines are not what we think of a parallel. They curve away from or toward each other. Said another way, in Euclidean geometry parallel lines can never cross. In non-Euclidean geometry they can.


Is it true that the sum of three angles of any triangle is 180 in non euclidean geometry?

No. Non-Euclidean geometries usually start with the axiom that Euclid's parallel postulate is not true. This postulate can be shown to be equivalent to the statement that the internal angles of a traingle sum to 180 degrees. Thus, non-Euclidean geometries are based on the proposition that is equivalent to saying that the angles do not add up to 180 degrees.


Will parallel lines intersect?

In Euclidean space, never. But they can in non-Euclidean geometries.


Are parallel lines coplanar?

In Euclidean geometry, parallel line are alwayscoplanar.


What is another name for the parallel postulate?

Playfair Axiom


Another name for the Playfair Axiom?

parallel postulate