That set of requirements narrows it down to only
an infinite number of possible unique triangles.
-- One angle is 60°. That leaves 120° for the sum of the other two.
-- One angle is obtuse. It can be anything more than 90°
and less than 120°.
-- And the third angle gets whatever is left.
-- If you don't mind fractional or decimal parts of degrees, then we
already have an infinite number of possible combinations of angles.
_____________________________________
Every possible combination of angles defines a unique set of
RATIOs among the sides.
But for EVERY unique set of RATIOs, there are an infinite number of possible unique triangles that are SIMILAR to each other.
Example:
If the angles determine that the sides must be in the ratio of 1:2:3,
then the triangle can have sides of
1, 2, and 3
2, 4, and 6
3, 6, and 9
4, 8, and 12
5, 10, and 15
6, 12, and 18
7, 14, and 21
8, 16, and 24
9, 18, and 27
10, 20, and 30
.
.
etc.
These all have the SAME set of ANGLES, and the same RATIO
among the sides, but they're all different unique triangles.