1. **Stating the problem:** We have two adjacent triangles sharing a vertex. One triangle has two angles marked as $x^\circ$, and the adjacent triangle has angles $43^\circ$ and $57^\circ$. We need to find the value of $x$.\n\n2. **Using the triangle angle sum rule:** The sum of the interior angles of any triangle is $180^\circ$.\n\n3. **Analyzing the triangle with angles $43^\circ$ and $57^\circ$:**\nThe third angle in this triangle is $$180^\circ - 43^\circ - 57^\circ = 80^\circ.$$\n\n4. **Considering the adjacent triangle with two angles $x^\circ$:**\nSince the two triangles share a vertex, the angle adjacent to the $80^\circ$ angle must be equal to $80^\circ$ (they are vertical angles).\n\n5. **Sum of angles in the triangle with two $x^\circ$ angles:**\n$$x^\circ + x^\circ + 80^\circ = 180^\circ.$$\n\n6. **Simplify and solve for $x$:**\n$$2x + 80 = 180$$\n$$2x = 180 - 80$$\n$$2x = 100$$\n$$x = \frac{100}{2}$$\n$$x = 50.$$\n\n**Final answer:** $x = 50^\circ$.