Since the two angles are adjacent and lie on a straight line, they sum up to 180 degrees. Therefore, we can set up the equation:
\[ (5x - 28) + (8x) = 180 \]
Now, combine the like terms:
\[ 5x - 28 + 8x = 180 \]
This simplifies to:
\[ 13x - 28 = 180 \]
Next, add 28 to both sides of the equation:
\[ 13x = 208 \]
Now, divide both sides by 13 to find \( x \):
\[ x = 16 \]
Now that we have the value of \( x \), we can substitute it back into the expression for the second angle:
\[ 8x = 8(16) = 128 \]
Therefore, the measure of the second angle is \( \boxed{128} \) degrees.