text: At 4:30, the longer hand will be pointing directly down.  The shorter hand will be 1/2 between the 4 and the 5.  If you equally divide the right half of the clock into 12 sectors, the number of sectors between the shorter hand and the longer hand is 3.  Given that 12/12 is 180 degrees, the angle between the two hands is 3/12 of 180 degrees or 45 degrees.