In a rhombus, the difference of the measures of the two angles between a side and the diagonals is 32°. What are the measures of the angles of the rhombus?

Imagine the angles between a side and the diagonals of a rhombus.  They are part of a right triangle, with the hypotenuse being the side of the rhombus and the legs being part of the diagonals.

The sum of the angles is a+b+90=180, so a+b=90.
We know that a-b=32, so a=32+b.  Let's plug that into the equation above:
32+b+b=90, so 32+2b=90, so 2b=58, and b=29.
We can back solve for a: a+29=90, so a=61.

The measure of the small angles is 29 and 61.  However, these are not the angles of the rhombus.  We must double them: the angles of the rhombus are 122 and 58.