1. **Problem Statement:** Find the angle $X$ in the given quadrilateral with the described angles and diagonals. 2. **Given Angles:** - Bottom-left vertex angles: $56^\circ$ and $30^\circ$ - Top-right vertex angle: $56^\circ$ - Bottom-right vertex angle: $43^\circ$ - Angle $X$ is between the diagonal and the right edge near the center. 3. **Step 1: Calculate the bottom-left vertex total angle.** The bottom-left vertex angle is split into two parts: $56^\circ + 30^\circ = 86^\circ$. 4. **Step 2: Use the fact that the sum of interior angles in a quadrilateral is $360^\circ$.** Let the top-left vertex angle be $A$. $$A + 56^\circ + 43^\circ + 86^\circ = 360^\circ$$ 5. **Step 3: Solve for $A$.** $$A = 360^\circ - (56^\circ + 43^\circ + 86^\circ) = 360^\circ - 185^\circ = 175^\circ$$ 6. **Step 4: Analyze the diagonals and angles formed.** The diagonals intersect, creating vertical angles and supplementary angles. 7. **Step 5: Use the angle at the top-right vertex ($56^\circ$) and bottom-right vertex ($43^\circ$) to find angle $X$.** Since angle $X$ is between the diagonal and the right edge near the center, it is supplementary to the $43^\circ$ angle at the bottom-right vertex. 8. **Step 6: Calculate angle $X$.** $$X = 180^\circ - 43^\circ = 137^\circ$$ **Final answer:** $$\boxed{137^\circ}$$