SOLUTION: A wheel of radius 10 cm rests against a step 5 cm high as shown. the wheel is rotated about Q by pushing until its center O is directly above Q. What is the angle OQ has now been r

That figure they gave you is not to scale to confuse your thinking,
even if you draw the circle in the new position.

So let's draw it to scale. The wheel is pushed up to the position in
red below:

 

Now let's draw in some horizontal and vertical lines,
plus 1 radius of the circle in the new position:



By drawing in those 3 congruent rectangles with height 5 cm and diagonal 10 cm,
we see that the circle has moved up 5 cm. and to the left the same as the
length of the bottom side of the 3 rectangles. Now triangle OPQ is equilateral,
so the angle we want is the angle OQP, and since triangle OPQ is equilateral,
angle OQP is 60o.  There actually is nothing to calculate.  All we
needed to do was to draw the circle to scale in the new position and draw those
3 congruent rectangles.

Edwin