1. **State the problem:** We need to find values of $x$ and $y$ such that quadrilateral $ABCD$ is a parallelogram. 2. **Key property:** In a parallelogram, the diagonals bisect each other. This means the two segments of each diagonal are equal. 3. **Set up equations for diagonal $AC$:** $$4x - 2 = x + 28$$ 4. **Solve for $x$:** $$4x - 2 = x + 28$$ $$4x - \cancel{2} = x + 28$$ $$4x - x = 28 + 2$$ $$3x = 30$$ $$x = \frac{30}{3} = 10$$ 5. **Set up equations for diagonal $DB$:** $$4y - 7 = y + 14$$ 6. **Solve for $y$:** $$4y - 7 = y + 14$$ $$4y - \cancel{7} = y + 14$$ $$4y - y = 14 + 7$$ $$3y = 21$$ $$y = \frac{21}{3} = 7$$ 7. **Conclusion:** For $ABCD$ to be a parallelogram, $x=10$ and $y=7$. **Final answer:** $x=10$, $y=7$