To solve for \( m\angle BCD \), we first set up the equation based on the information given about supplementary angles. Supplementary angles sum to \( 180^\circ \).
Given:
- \( m\angle ACB = (x + 28)^\circ \)
- \( m\angle BCD = (x + 48)^\circ \)
Since they are supplementary, we can write the equation:
\[ (x + 28) + (x + 48) = 180 \]
Now, combine like terms:
\[ 2x + 76 = 180 \]
Next, subtract \( 76 \) 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 \( m\angle BCD \):
\[ m\angle BCD = x + 48 = 52 + 48 = 100 \]
Therefore,
\[ m\angle BCD = 100^\circ \]
The answer is \( \boxed{100^\circ} \).