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Is hyperbolic parallel postulate a postulate of Euclid?
No, the hyperbolic parallel postulate is not one of Euclid's original five postulates. Euclid's fifth postulate, known as the parallel postulate, states that given a line and a point not on that line, there is exactly one line parallel to the original line that passes through the point. Hyperbolic geometry arises from modifying this postulate, allowing for multiple parallel lines through the given point, leading to a different set of geometric principles.
Which conjecture justifies the construction of a line parallel to a given line through a given point?
Euclid's parallel postulate.
How did you know that the sum of the angles in a triangle is 180?
It is proven by a theorem (which relies on Euclid's parallel postulate).
Why does a triangle have 180 degree?
It is a consequence of Euclid's parallel postulate. In fact, in some versions, the statement that "a plane triangle has interior angles that sum to 180 degrees" replaces the parallel postulate.
If there is a line and a point not on the line then there is exactly lines trough the point parallel to the given line?
This is Euclid's fifth postulate, also known as the Parallel Postulate. It is quite possible to construct consistent systems of geometry where this postulate is negated - either many parallel lines or none.
Why the measure of exterior angles is equal to the sum of the measures of interior opposite angles?
That is only true of triangles and is a consequence of the parallel postulate. In fact it is an alternative way of stating Euclid's parallel postulate.
Through a point not on the line exactly one line can be drawn parallel to the?
... given line. This is one version of Euclid's fifth postulate, also known as the Parallel Postulate. It is quite possible to construct consistent systems of geometry where this postulate is negated - either many parallel lines or none.
What is another name for the Playfair Axiom?
Another name for the Playfair Axiom is the Euclid's Parallel Postulate. It states that given a line and a point not on that line, there is exactly one line parallel to the given line passing through the given point.
What triangle sum says that the 3 angles in a triangle always equals 180 degrees?
It is a consequence of Euclid's parallel postulate.
Who restated Euclid's 5th postulate?
Probably the best known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish mathematician John Playfair, which states:In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.
Does fifth postulate of Euclid imply the existence of parallel lines?
Yes.According to Euclid's 5th postulate, when n line falls on l and m and if, producing line l and m further will meet in the side of ∠1 and ∠2 which is less thanIfThe lines l and m neither meet at the side of ∠1 and ∠2 nor at the side of ∠3 and ∠4. This means that the lines l and m will never intersect each other. Therefore, it can be said that the lines are parallel.
Does Euclid's fifth postulate imply the existence of parallel lines?
Yes.According to Euclid's 5th postulate, when n line falls on l and m and if, producing line l and m further will meet in the side of ∠1 and ∠2 which is less thanIfThe lines l and m neither meet at the side of ∠1 and ∠2 nor at the side of ∠3 and ∠4. This means that the lines l and m will never intersect each other. Therefore, it can be said that the lines are parallel.
