Question

You can only draw one unique isosceles triangle that contains an angle of 55°?

Answer 1

-- If the 55° angle is one of the two equal angles, then
the third angle is 70° .

-- If the 55° angle is the third angle, then each of the two
equal angles is 62.5° .

-- For either of these cases, there are an infinite number
of possible sets of side-lengths.

The statement in the question does not hold water.