How many unique triangles can be made when one angle measures 90° and another angle is half that measure?

Infinite number        

Step-by-step explanation:

we know that                      

If one angle is 90°, and another half of that, then it must be 45° and so the third angle is also 45°. (The sum of the internal angles of a triangle must be equal to 180 degrees)

Remember that                

If two triangles are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent

therefore                    

There is an infinite number of unique triangle with these angles.