1. **State the problem:** We are given a quadrilateral with $AB \parallel DC$, and three angles at point $A$: $\angle 1 = 112^\circ$, $\angle 2 = 4x$, and $\angle 3 = 3x + 12$. We need to find the measure of $\angle 3$.\n\n2. **Identify relationships:** Since $AB \parallel DC$, angles around point $A$ that lie on a straight line sum to $180^\circ$. The angles $\angle 1$, $\angle 2$, and $\angle 3$ together form a straight angle at $A$.\n\n3. **Write the equation:** The sum of the three angles is $180^\circ$, so\n$$112 + 4x + (3x + 12) = 180$$\n\n4. **Simplify the equation:** Combine like terms:\n$$112 + 4x + 3x + 12 = 180$$\n$$112 + 7x + 12 = 180$$\n$$7x + 124 = 180$$\n\n5. **Isolate $x$:** Subtract 124 from both sides:\n$$7x + \cancel{124} - \cancel{124} = 180 - 124$$\n$$7x = 56$$\n\n6. **Solve for $x$:** Divide both sides by 7:\n$$\frac{7x}{\cancel{7}} = \frac{56}{\cancel{7}}$$\n$$x = 8$$\n\n7. **Find $\angle 3$:** Substitute $x=8$ into $\angle 3 = 3x + 12$:\n$$\angle 3 = 3(8) + 12 = 24 + 12 = 36^\circ$$\n\n**Final answer:**\n$$\boxed{36^\circ}$$