It is known that in Hyperbolic Geometry angle sum of a triangle is strictly less than $\pi.$
Can we find a hyperbolic triangle with each angle is zero?
If it is so, is there any characterization of such triangles?
If not, is there any geometry in which "triangle with each angle is zero" exists?
As Will Jagy pointed out, the answer is yes and these are called ideal triangles. Perhaps the simplest explicit example is given by the set
$$T = \{(x,y) \colon |x|<1, \ y>\sqrt{1-x^2} \}$$
using the upper-half model of the hyperbolic plane. This is the triangle pictured on the left in Wikipedia illustration: