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

In a right-triangle, if either of the other two angles are "half that measure", or half of 90°, that means 45°, then the last angle will also have to be a 45° one too, and the sides coming from the 90° angle, will be equal twins, and therefore they can only make up one type-length hypotenuse, and due to that, you can only make that one triangle.  Check picture below.