Since the two angles are adjacent on a straight line, they are supplementary, meaning they add up to \(180^\circ\).
Given the two angles:
- \( 5x - 28 \)
- \( 8x \)
We can set up the equation:
\[ (5x - 28) + 8x = 180 \]
Now, combine like terms:
\[ 13x - 28 = 180 \]
Next, add \(28\) to both sides:
\[ 13x = 180 + 28 \] \[ 13x = 208 \]
Now, divide both sides by \(13\):
\[ x = \frac{208}{13} \] \[ x = 16 \]
Now, we can find the measure of the second angle, which is \(8x\):
\[ 8x = 8(16) = 128 \]
Thus, the measure of the second angle is \( \boxed{128} \).