Since angles \( \angle ACB \) and \( \angle BCD \) are supplementary, their measures add up to \( 180^\circ \). Therefore, we can set up the equation:
\[ \angle ACB + \angle BCD = 180^\circ \]
Substituting the expressions for the angles:
\[ (x + 28) + (x + 48) = 180 \]
Combining like terms gives:
\[ 2x + 76 = 180 \]
Next, we solve for \( x \) by subtracting \( 76 \) from both sides:
\[ 2x = 180 - 76 \] \[ 2x = 104 \]
Now, divide by \( 2 \):
\[ x = 52 \]
Now that we have the value of \( x \), we can find the measure of \( \angle BCD \):
\[ \angle BCD = x + 48 = 52 + 48 = 100^\circ \]
So, the measure of \( \angle BCD \) is
\[ \boxed{100^\circ} \]
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