To find the measure of angle \( m∠ACB \), we know that supplementary angles add up to \( 180^\circ \). In this case, we have:
\[ m∠ACB + m∠BCD = 180^\circ \]
Substituting the expressions for the angles:
\[ 4x + (6x + 50) = 180 \]
Now, combine like terms:
\[ 4x + 6x + 50 = 180 \] \[ 10x + 50 = 180 \]
Next, subtract \( 50 \) from both sides:
\[ 10x = 130 \]
Now, divide by \( 10 \):
\[ x = 13 \]
Now we can find \( m∠ACB \):
\[ m∠ACB = 4x = 4(13) = 52^\circ \]
Thus, the measure of \( ∠ACB \) is
\[ \boxed{52^\circ} \]