Angle X Value 6421E4

1. **State the problem:** We have a construction with black rods all equal in length and white rods all equal in length. An angle of $40^\circ$ is given, and we need to find the angle $x$ marked in the figure. 2. **Analyze the figure and given information:** The construction forms a triangle-like shape with an isosceles triangle on top. Since all black rods are equal and all white rods are equal, the triangle formed by black rods is isosceles. 3. **Use properties of isosceles triangles:** In an isosceles triangle, the base angles are equal. Given one angle is $40^\circ$, the other base angle is also $40^\circ$. 4. **Calculate the vertex angle of the isosceles triangle:** The sum of angles in a triangle is $180^\circ$, so the vertex angle is $$180^\circ - 40^\circ - 40^\circ = 100^\circ.$$ 5. **Identify angle $x$:** Angle $x$ is adjacent to the $40^\circ$ angle and formed between a black rod and a white rod. Since the rods are equal in length and the construction is symmetric, angle $x$ corresponds to the vertex angle of the smaller triangle formed by the rods. 6. **Calculate angle $x$:** Using the linear pair property, angles on a straight line sum to $180^\circ$. Since the adjacent angle is $40^\circ$, angle $x$ is $$x = 180^\circ - 40^\circ = 140^\circ.$$ **Final answer:** $$\boxed{140^\circ}$$