Step 1 :Given that triangles Y M Z and Q M P are identical, they have the same side lengths.
Step 2 :We know that the length of YZ is 13, and the lengths of QM and PM are both 15.
Step 3 :Therefore, the remaining side lengths of the triangles are also 13 and 15.
Step 4 :So, MY = 13, ZM = 13, PQ = 15, MP = 15.
Step 5 :Final Answer: \(\boxed{MY = 13, ZM = 13, PQ = 15, MP = 15}\)