To determine the measure of angle \( \angle BCD \), we can use the fact that supplementary angles add up to \( 180^\circ \).
Given:
- \( m\angle ACB = (x + 28)^\circ \)
- \( m\angle BCD = (x + 48)^\circ \)
Because these angles are supplementary, we set up the following equation:
\[ (x + 28) + (x + 48) = 180 \]
Now, let's combine like terms:
\[ 2x + 76 = 180 \]
Next, we will isolate \( 2x \) by subtracting \( 76 \) from both sides:
\[ 2x = 180 - 76 \] \[ 2x = 104 \]
Now, divide both sides by \( 2 \):
\[ x = 52 \]
Now that we have the value of \( x \), we can find the measure of \( \angle BCD \):
\[ m\angle BCD = x + 48 = 52 + 48 = 100^\circ \]
Thus, the measure of \( \angle BCD \) is:
\[ \boxed{100^\circ} \]