What kind of transformation is illustrated in this figure? dilation translation rotation reflection triangle d e f and another triangle d-prime e-prime f-prime is of the same size, shape but turned a different angle

Without the drawing, it's impossible to say for sure. However, it is definitely not a dilation because of the "same size" fact stated. If a dilation did occur, it would be smaller or larger than the original. This is also known as scaling.

So its either a translation, rotation or reflection. I don't think it's a translation because you mention how its "turned a different angle" so that suggests a rotation perhaps. Though that doesn't fully rule out a reflection. If you can please post the diagram so I can have a look. 

How much are you paying ?

Hope i can you!