[center][/center][center]Hyperbolic geometry is a non-Euclidean geometry is also known as Lobachevsky-Bolyai-Gauss geometry, due to its cofounders. This geometry satisfies all of Euclid's postulates [i]except for[/i] the fifth postulate, the parallel postulate. [br][br]Euclid's Fifth Postulate Reads: [color=#1155cc]"[i]If[/i][/color][i][color=#1155cc] a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."[/color][br][br][/i]This postulate is modified for Hyperbolic Geometry to read: [color=#38761d]"[i]For any infinite straight [url=https://mathworld.wolfram.com/Line.html]l[/url]ine L and any point P not on it, there are many other infinitely extending straight [url=https://mathworld.wolfram.com/Line.html]l[/url]ines that pass through P and which do not intersect L.[/i][/color][/center]
Since the postulate itself can be difficult to understand just from reading it, interact with the display below to understand what Euclid's 5th Postulate means
Do the lines intersect on that side?
Try constructing these lines in the applet below[br][list=1][*]PQ[/*][*]PR[/*][*]PS[/*][*]PT[/*][/list]
Did any of these lines intersect the given line L?
How many lines do you think you could construct that go through point P, but do not intersect with line L? Why?