1. **State the problem:** We need to find the sum of the angles $f^\circ + g^\circ$ in the given quadrilateral figure. 2. **Recall the angle sum property of quadrilaterals:** The sum of the interior angles of any quadrilateral is always $$360^\circ$$. 3. **Identify the known angles:** From the problem, the angles given are $a^\circ$, $60^\circ$, $c^\circ$, $d^\circ$, $e^\circ$, $50^\circ$, $f^\circ$, $g^\circ$, $h^\circ$, $i^\circ$, $k^\circ$, $110^\circ$, $180^\circ$, $120^\circ$, and $360^\circ$. However, the problem focuses on the quadrilateral with vertices having angles $a^\circ$, $60^\circ$, $c^\circ$ at the top-left vertex; $d^\circ$, $e^\circ$ at the top-right vertex; $f^\circ$, $g^\circ$ at the bottom-left vertex; and $50^\circ$, $r^\circ$, $k^\circ$ at the bottom-right vertex. 4. **Use the sum of interior angles:** Since the quadrilateral's interior angles sum to $360^\circ$, we write: $$a + 60 + c + d + e + f + g + 50 + r + k = 360$$ 5. **Focus on the bottom-left vertex:** The problem asks for $f + g$. Since the other angles are not numerically specified or related, and the figure is complex, the key is to recognize that the sum of the interior angles at the bottom-left vertex ($f + g$) plus the other angles must total $360^\circ$. 6. **Use the exterior angle property:** The sum of the exterior angles of any polygon is $360^\circ$. If $f$ and $g$ are interior angles at the bottom-left vertex, their sum can be found by subtracting the other known angles from $360^\circ$. 7. **Conclusion:** Without additional numeric relationships or values, the sum $f^\circ + g^\circ$ cannot be determined exactly from the given information. **Final answer:** Insufficient information to determine $f^\circ + g^\circ$.