To find the measure of the second angle, we can set up the equation 5x - 28 + 8x = 180, since the sum of the two adjacent angles on a straight line is 180 degrees.
Combining like terms, we have 13x - 28 = 180.
Adding 28 to both sides, we get 13x = 208.
Dividing both sides by 13, we find x = 16.
Substituting x = 16 into 8x, we get 8(16) = 128.
Therefore, the measure of the second angle is 128 degrees.