1. **Problem statement:** We have an equilateral triangle $\triangle ABC$ and a square $DEFG$ inscribed inside it such that $D$ lies on $AB$, $E$ on $BC$, and $F$ on $AC$. Lines $ADB$, $AFC$, and $BEC$ are straight. We need to find the sum of angles $x$ and $y$ marked at vertices $D$ and $F$ respectively. 2. **Key properties:** - In an equilateral triangle, all angles are $60^\circ$. - A square has all angles equal to $90^\circ$. - Since $DEFG$ is a square, angles at $D$, $E$, $F$, and $G$ are right angles. 3. **Analyze angle $x$ at $D$:** - $D$ lies on line $AB$. - The square angle at $D$ is $90^\circ$. - The line $ADB$ is straight, so angle $ADB = 180^\circ$. - The angle $x$ is the angle between $AD$ and $DB$ inside the triangle, adjacent to the square's right angle. 4. **Analyze angle $y$ at $F$:** - $F$ lies on line $AC$. - The square angle at $F$ is $90^\circ$. - The line $AFC$ is straight, so angle $AFC = 180^\circ$. - The angle $y$ is the angle between $AF$ and $FC$ inside the triangle, adjacent to the square's right angle. 5. **Use the fact that $\triangle ABC$ is equilateral:** - Each angle of $\triangle ABC$ is $60^\circ$. - At vertex $A$, the angle between $AB$ and $AC$ is $60^\circ$. 6. **Relate angles at $D$ and $F$ to the triangle's angles:** - Since $D$ is on $AB$ and $F$ is on $AC$, and $DEFG$ is a square, the angles $x$ and $y$ complement the right angles of the square to fit inside the $60^\circ$ angles of the triangle. 7. **Sum of $x$ and $y$:** - The sum of angles around point $A$ inside the triangle is $60^\circ$. - The square's right angles at $D$ and $F$ take $90^\circ$ each, but since $D$ and $F$ lie on the sides, the angles $x$ and $y$ are the supplements inside the triangle. 8. **Conclusion:** - The sum $x + y$ equals the angle at vertex $A$ of the equilateral triangle, which is $60^\circ$. **Final answer:** $$x + y = 60^\circ$$