To the nearest tenth of a degree, find the sizes of the acute angles in the 7-24-25 right
triangle and in the 8-15-17 right triangle. this information then allows you to calculate
the sizes of all the angles in the 25-51-52 triangle.

First of all, just to avoid being snookered by a trick question, we should verify that these are really right triangles:

7² + 24² really is 25² , and 8² + 15² really is 17² , so we're OK there.

In the first one:
sin(one acute angle) = 7/25 = 0.28
the angle = sin⁻¹ (0.28)  =  16.26°
the other acute angle = (90° - 16.26°)  = 73.74° 

In the second one:
sin(one acute angle) = 8/17 = 0.4706...
the angle = sin⁻¹  (0.4706...)  =  28.07°
the other acute angle =  (90° - 28.07°) = 61.93°

I'm sorry, but just now, I don't know how to do the 
third triangle in the question.

A’b’c’ are the coordinates to the new location of the transformed triangle abc. find the distance between the coordinates points to use for your calculations.