1. **State the problem:** We are given a quadrilateral with $AB \parallel DC$, and angles at vertex $A$ labeled as $\angle 1 = 112^\circ$, $\angle 2 = 4x$, and $\angle 3 = 3x + 12$. We need to find the measure of $\angle D$. 2. **Identify relationships and formulas:** Since $AB \parallel DC$, alternate interior angles and corresponding angles formed by transversals are related. Also, the angles around point $A$ sum to $360^\circ$ because they form a full circle around that vertex. The angles at $A$ are adjacent and together with the straight lines, so we can use angle sum properties. 3. **Sum of angles at vertex $A$:** The three angles $\angle 1$, $\angle 2$, and $\angle 3$ are adjacent and form a straight angle with the line $AB$, so their sum is $180^\circ$. Thus, $$\angle 1 + \angle 2 + \angle 3 = 180^\circ$$ Substitute the given values: $$112 + 4x + (3x + 12) = 180$$ 4. **Simplify and solve for $x$:** $$112 + 4x + 3x + 12 = 180$$ $$112 + 12 + 7x = 180$$ $$124 + 7x = 180$$ Subtract 124 from both sides: $$7x = 180 - 124$$ $$7x = 56$$ Divide both sides by 7: $$x = \frac{\cancel{7}56}{\cancel{7}} = 8$$ 5. **Find $\angle 2$ and $\angle 3$ using $x=8$:** $$\angle 2 = 4x = 4 \times 8 = 32^\circ$$ $$\angle 3 = 3x + 12 = 3 \times 8 + 12 = 24 + 12 = 36^\circ$$ 6. **Find $\angle D$:** Since $AB \parallel DC$, $\angle D$ and $\angle 1$ are alternate interior angles formed by transversal $AD$. Therefore, $$m\angle D = m\angle 1 = 112^\circ$$ **Final answer:** $$\boxed{112^\circ}$$