To find the measure of angle \( \angle BCD \), we start with the fact that the angles \( \angle ACB \) and \( \angle BCD \) are supplementary, meaning their measures add up to \( 180^\circ \).
Given the measures: \[ \angle ACB = (x + 28)^\circ \] \[ \angle BCD = (x + 48)^\circ \]
We can set up the following equation: \[ (x + 28) + (x + 48) = 180 \]
Now, we simplify this equation: \[ 2x + 76 = 180 \]
Next, we subtract \( 76 \) from both sides: \[ 2x = 180 - 76 \] \[ 2x = 104 \]
Now, we divide both sides by \( 2 \): \[ x = 52 \]
Now that we have the value of \( x \), we can find the measure of \( \angle BCD \): \[ \angle BCD = x + 48 = 52 + 48 = 100 \]
Thus, the measure of \( \angle BCD \) is: \[ \boxed{100 \text{ degrees}} \]