In this triangle, we have three angles marked with values in terms of $x$:

1. $40^\circ$ (top angle)
2. $2x^\circ$ (left angle)
3. $x + 100^\circ$ (bottom angle)
4. $x^\circ$ (right angle)

Since this is a triangle, the sum of the angles must equal $180^\circ$. Therefore, we can set up the following equation:

$40 + 2x + (x + 100) + x = 180$

Now, we combine like terms:

$40 + 2x + x + 100 + x = 180$ $40 + 100 + 4x = 180$ $140 + 4x = 180$

Subtracting 140 from both sides:

$4x = 40$

Now, divide by 4:

$x = 10$

So, the value of $x$ is $10^\circ$.

Would you like further details or have any questions?

Here are some related questions:

1. What is the importance of the angle sum property in a triangle?
2. How can this knowledge be applied to solve real-life geometry problems?
3. What would the triangle look like if $x$ had a different value?
4. How would the solution change if the triangle had a different configuration?
5. What other geometric properties can be used to find unknown values in polygons?

Tip: Always double-check your equation setup when summing angles in a polygon to avoid simple mistakes.