[left]Hyperbolic Geometry fails to meet the criteria for the 5th postulate, so the modified 5th postulate reads:[br][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][/left]

[left]The photo above also depicts what we learned about the interior angles of a Hyperbolic triangle[br][color=#38761d]The sum of the interior angles is always less than 180 degrees[/color][br][/left]

[left]Another thing we learned about Hyperbolic triangles is that[br][math]a^2+b^2[/math]<[math]c^2[/math][/left]

[left]There are some architectural structures designed using Hyperbolic Geometry[/left]

Designed by Eero Saarinen, the Dulles airport is the shape of a hyperbolic paraboloid.

[left]The hyperboloid is the design standard for all nuclear cooling towers and some coal-fired power plants. [br]They are built this way because the broad base allows for greater area to encourage evaporation, then narrows to increase air flow velocity. It then widens slightly to aid in mixing the moisture laden air into the atmosphere.[/left]

Take a look at the pictures below. All are examples of objects with Hyperbolic shape