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Exactly one line can be drawn through any point not on a given line parallel to the given line in a plane Euclids 5th states 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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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.
If there sides of a triangle are respectively equal to the three sides of another triangle the two triangles are we call this as axiom?
It is not an axiom, but a theorem.
Is there any number that 1 times itself does not equal itself?
No, because 1 times any number is an axiom, or law, of math; The identity axiom of multiplication, that states any number that is a real number multiplied by 1 equals itself. ex. a x 1 = a, a = 5 5 x 1 = 5 Results will be the same for any real number.
Is the continuum hypothesis true?
Continuum hypothesis was proven, with an proving method called "forcing", to be undecidable under commonly accepted axioms of the set theory. This means that neither continuum hypothesis nor it's negation follows from this axioms just like one axiom (or it's negation) in some consistent axiomatic system does not follow from other axioms. Therefore, continuum hypothesis or it's negation could be added as an additional axiom to existing commonly accepted axioms of the set theory.
What is Euclid's Axiom?
Euclid posited five axioms, statements whose truth supposedly does not require a proof, as the foundation of his work, the Elements. These still hold for plane geometry, but do not hold in the higher non-euclidean systems. The five axioms Euclid proposed are;Any two points can be connected by one, and only one, straight line.Any line segment can be extended infinitelyFor any point, and a line emerging from it, a circle can be drawn where the point is the centre and the line is the radius.All right angles are equalGiven a line, and a point not on the line, there is only line that goes through the point that does not meet the other line. (basically, there is only one parallel to any given line)This last point is controversial as it has been argued effectively that this is not in fact self evident. In fact, ignoring the fifth axiom was the starting point for many Non-Euclidean geometries. For this reason, it is probably this which is best known as Euclid's Axiom.
