1. **State the problem:** We need to find the value of angle $x$ in a triangle where two exterior angles are given: $120^\circ$ at vertex A and $112^\circ$ at vertex C. 2. **Recall the exterior angle theorem:** The exterior angle of a triangle is equal to the sum of the two opposite interior angles. 3. **Apply the theorem at vertex A:** The exterior angle at A is $120^\circ$, so $$120 = x + \angle C$$ 4. **Apply the theorem at vertex C:** The exterior angle at C is $112^\circ$, so $$112 = x + \angle A$$ 5. **Use the triangle angle sum:** The sum of interior angles in a triangle is $180^\circ$, so $$\angle A + x + \angle C = 180$$ 6. **From step 3, express $\angle C$:** $$\angle C = 120 - x$$ 7. **From step 4, express $\angle A$:** $$\angle A = 112 - x$$ 8. **Substitute $\angle A$ and $\angle C$ into the triangle sum:** $$ (112 - x) + x + (120 - x) = 180 $$ 9. **Simplify the equation:** $$ 112 - x + x + 120 - x = 180 $$ $$ 112 + 120 - x = 180 $$ $$ 232 - x = 180 $$ 10. **Solve for $x$:** $$ 232 - x = 180 $$ $$ \cancel{232} - x = \cancel{180} $$ $$ -x = 180 - 232 $$ $$ -x = -52 $$ $$ x = 52 $$ **Final answer:** $$\boxed{52^\circ}$$