1. **Stating the problem:** We are given angles $x^\circ$, $w^\circ$, $y^\circ$ with the condition $BA = BD$ and the ratio $x : y = 2 : 1$. We need to find the relationship between these angles. 2. **Understanding the problem:** Since $BA = BD$, triangle $BAD$ is isosceles with $BA = BD$. This implies the base angles opposite these sides are equal. 3. **Using the isosceles triangle property:** Let the vertex angle be $w^\circ$ and the base angles be $x^\circ$ and $y^\circ$. Since $x : y = 2 : 1$, and the base angles are equal, we set $x = y$. 4. **Contradiction check:** The ratio $x : y = 2 : 1$ contradicts the equality of base angles in an isosceles triangle. Therefore, the ratio applies to different angles, not the base angles. 5. **Sum of angles in triangle:** The sum of angles in triangle $BAD$ is $$x + w + y = 180^\circ$$ 6. **Expressing $x$ in terms of $y$:** Given $x : y = 2 : 1$, we write $$x = 2y$$ 7. **Substitute into angle sum:** Substitute $x = 2y$ into the sum: $$2y + w + y = 180^\circ$$ $$3y + w = 180^\circ$$ 8. **Express $w$ in terms of $y$:** $$w = 180^\circ - 3y$$ 9. **Conclusion:** The angles satisfy $$x = 2y$$ and $$w = 180^\circ - 3y$$ with $BA = BD$ implying $x = y$ is not possible here, so the ratio applies to other angles. **Final answer:** $$x = 2y, \quad w = 180^\circ - 3y$$