1. **State the problem:** We are given a quadrilateral with $AB \parallel DC$, and three angles: $m\angle 1 = 112^\circ$, $m\angle 2 = 4x$, and $m\angle 3 = 3x + 12$. We need to find the measure of $\angle 2$. 2. **Identify relationships:** Since $AB \parallel DC$, angles formed by a transversal have special relationships. Here, $\angle 1$ and $\angle 3$ are consecutive interior angles, so their measures add up to $180^\circ$. 3. **Write the equation:** $$ 112 + (3x + 12) = 180 $$ 4. **Simplify and solve for $x$:** $$ 112 + 3x + 12 = 180 $$ $$ 3x + 124 = 180 $$ $$ 3x = 180 - 124 $$ $$ 3x = 56 $$ $$ x = \frac{56}{3} $$ 5. **Use $x$ to find $m\angle 2$:** $$ m\angle 2 = 4x = 4 \times \frac{56}{3} = \frac{224}{3} \approx 74.67^\circ $$ **Final answer:** $$ m\angle 2 = \frac{224}{3}^\circ \approx 74.67^\circ $$