Question

How many triangles can be constructed with angles measuring 120°, 40°, and 10°? one more than one none

Answer 1

I might be wrong but i believe you can make more than one with those 3 angles. the 120 and 40 can make one and if angled right can make 2 and the 10 deg on one of the other angles lines would give you a 3rd.
 

Answer 2

The interior angles of a triangle must sum to 180 degrees.

$\rm 120 + 40 + 10\ne180$

These angles do not sum to the proper amount. No triangle can be constructed using all three angles.