How do you find the value of equal angles y and x?

Because we know the triangles are similar and the angle of D = angle of A, that means that the angle of B = angle of E and angle of C = angle of F. Therefore, angle of C = 78 degrees and angle of E = 64 degrees. The angles of a triangle sum to 180 degrees, so we have the equation: x + 64 + 78 = 180. Solving for x gives you x = 38 degrees, which is the same as angle y. 

They are similar
So
B = E
C = F
x° = y°

x = y = 180° - 64° - 78° = 38°