If an object is reflected twice over parallel lines, and corresponding points of the original and final translation image are 30 centimeters apart, how far apart are the parallel lines?

Think about it this way, you have the object touching one of the parallel lines, with point P of the object touching the first parallel line. Now the point on the opposite side of the object as point P, lets call is P', is touching the second parallel line. Next, reflecting across the second parallel line P' remains touching the second parallel line while P is moved first moved x units in order to make it touch the second parallel line, and then moved another x units so that is has been reflected, therefore it has been moved 2x units.

After having been moved 2x units the distance that P has traveled is 30,
so x = 15.