The answer will depend on whether the 3 values refer to the measures of the angles or the lengths of sides.
In the first case, they must be three positive numbers that add to exactly 180 degrees.
In the second case, the sum of any two of them must be greater than the third.
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If it has a 90 degree angle and two acute angles or complies with Pythagoras' theorem for a right angle triangle
The three vertices of the triangle uniquely determine a circle that circumscribes the triangle. The three sides of the triangle uniquely determine the circle that inscribes the triangle.
If it's not a right angled triangle and you don't have any of the angles but have the values of all three sides, then you need to use something called the Cosine Rule.
Pick three numbers. If the square of the largest number is equal to the sum of the squares of the other two, then the three numbers could represent the sides of a right triangle.
with the three sides you know it will be a triangle, it is a right triangle, it is not an equilateral triangle or an isosceles triangle... so it must be a right scalene triangle
Three non-collinear points do not determine a unique spherical triangle.