To find the measure of angle \( m\angle BCD \), we need to use the fact that angles \( \angle ACB \) and \( \angle BCD \) are supplementary. This means that their measures add up to \( 180^\circ \).
We have: \[ \angle ACB = x + 28 \] \[ \angle BCD = x + 48 \]
Setting up the equation based on the supplementary relationship: \[ (x + 28) + (x + 48) = 180 \]
Now we can combine like terms: \[ 2x + 76 = 180 \]
Next, we isolate \( 2x \) by subtracting \( 76 \) from both sides: \[ 2x = 180 - 76 \] \[ 2x = 104 \]
Now, we solve for \( x \) by dividing both sides by \( 2 \): \[ x = 52 \]
Now that we have \( x \), we can find \( m\angle BCD \): \[ m\angle BCD = x + 48 = 52 + 48 = 100 \]
Thus, the measure of \( m\angle BCD \) is \( \boxed{100} \).