In a cyclic quadrilateral, the sum of opposite angles is $180^\circ$. Given the opposite angles are $(3x+10)^\circ$ and $(4x+30)^\circ$, we set their sum equal to $180^\circ$: $(3x+10) + (4x+30) = 180$. Combining like terms, $7x + 40 = 180$. Subtracting $40$ from both sides, $7x = 140$. Dividing by $7$, $x = 20$. Now, substitute $x=20$ back into the expressions for the angles: First angle $= (3(20)+10)^\circ = (60+10)^\circ = 70^\circ$. Second angle $= (4(20)+30)^\circ = (80+30)^\circ = 110^\circ$. The measures of the angles are $70^\circ$ and $110^\circ$.