The law of cosines can be stated as
where the lowercase letters are sides of the triangle and the capital letters are the angles opposite the respective sides. (There are actually three different ways you can write this formula depending on which angle you have, so I chose the version with with A, for which I will use the 55° angle.)
So we can plug in the #s we have and solve for the unknown c.
At this point, it looks like a quadratic, so you should be able to take it home from here and get the missing lengths on the diagram you have which I can't see. Make sure to only use the positive result.