1. **Problem statement:** We have a trapezoid with two diagonals intersecting inside it, forming an angle $x$ at the intersection. Two angles near the intersection are given: $38^\circ$ (top left) and $43^\circ$ (bottom left). We need to find the value of $x$. 2. **Key idea:** When two lines intersect, the angles around the intersection point add up to $360^\circ$. Also, vertically opposite angles are equal. 3. **Step-by-step solution:** - The diagonals intersect forming four angles around the point. Two of these angles are given: $38^\circ$ and $43^\circ$. - The angle $x$ is adjacent to these two angles, and the fourth angle is vertically opposite to $x$, so it also equals $x$. - The sum of all four angles around the intersection is $360^\circ$: $$38^\circ + 43^\circ + x + x = 360^\circ$$ - Combine like terms: $$81^\circ + 2x = 360^\circ$$ - Subtract $81^\circ$ from both sides: $$2x = 360^\circ - 81^\circ$$ $$2x = 279^\circ$$ - Divide both sides by 2: $$x = \frac{279^\circ}{2}$$ - Show cancellation: $$x = \frac{\cancel{2}79^\circ}{\cancel{2}}$$ - Simplify: $$x = 139.5^\circ$$ 4. **Answer:** The angle $x$ formed by the diagonals at their intersection is $139.5^\circ$.