To solve for $x$, we use the properties of triangles and the relationships between the angles. Here's the step-by-step approach:

Step 1: Understanding the geometry

• The diagram shows a non-standard quadrilateral split into two triangles.
• The key property here is that the sum of angles in any triangle is $180^\circ$.
• We will apply this rule to both the triangles.

Step 2: Solve for the triangle at the top

In the top triangle: $\text{Sum of angles: } 40^\circ + (x+1)^\circ + z = 180^\circ$ $z = 180^\circ - 40^\circ - (x+1)^\circ = 139^\circ - x$

Step 3: Solve for the triangle at the bottom

In the bottom triangle: $\text{Sum of angles: } 2x + x + (z+100) = 180^\circ$ Substitute $z = 139^\circ - x$ into this equation: $2x + x + ((139 - x) + 100) = 180^\circ$ Simplify: $3x + 239 - x = 180$ $2x + 239 = 180$ $2x = 180 - 239$ $2x = -59$ $x = -29.5$

The value of $x$ is $-29.5$, which implies a possible error system