Two angles are supplementary, which means their sum is 180 degrees. Given the measures of the angles:
\[ m∠ACB = 4x° \] \[ m∠BCD = (6x + 50)° \]
We can set up the equation:
\[ 4x + (6x + 50) = 180 \]
Combine like terms:
\[ 10x + 50 = 180 \]
Now, subtract 50 from both sides:
\[ 10x = 130 \]
Next, divide both sides by 10:
\[ x = 13 \]
Now, we can find the measure of angle \( m∠ACB \):
\[ m∠ACB = 4x = 4(13) = 52° \]
Therefore, the measure of \( ∠ACB \) is:
\[ \boxed{52°} \]