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.

Ireally don’t understand your problem do you have a better picture of it