To find the measure of angle \( \angle BCD \), we know that angles \( \angle ACB \) and \( \angle BCD \) are supplementary, which means their measures add up to \( 180^\circ \).
Given the measures: \[ m\angle ACB = (x + 28)^\circ \] \[ m\angle BCD = (x + 48)^\circ \]
Since the angles are supplementary, we can write the equation: \[ (x + 28) + (x + 48) = 180 \]
Now, let's simplify this equation: \[ 2x + 76 = 180 \]
Next, we will isolate \( x \): \[ 2x = 180 - 76 \] \[ 2x = 104 \] \[ x = 52 \]
Now that we have the value of \( x \), we can find \( m\angle BCD \): \[ m\angle BCD = x + 48 = 52 + 48 = 100 \]
Thus, the measure of \( \angle BCD \) is \( \boxed{100^\circ} \).