A general problem solving idea that's good to have in your toolbox is "the whole is equal to the sum of its parts." Here, our parts are the angles ∠STC and ∠CTU, and they combine to make the whole, angle ∠STU. We can write this as the equation
m∠CTU + m∠STC = m∠STU
Using the values m∠CTU = 89° and m∠STU = 155°, this becomes
89° + m∠STC = 155°
The only value that makes this true is 66°, so we can conclude that m∠STC = 66°.