https://file.aiquickdraw.com/mathbot/file/1728996160009l6heyjjz.jpg

To solve for \( x \) in the given figure, we can break the problem down into different parts using the properties of triangles and angles.

### Step 1: Identify Key Angles and Triangles
- The large triangle at the bottom contains a right angle (denoted by the small square), and the angle at the top of the figure is \( 30^\circ \). This triangle also includes angle \( x^\circ \) and another angle adjacent to it, which we'll need to determine.
  
### Step 2: Apply the Exterior Angle Theorem
Notice that the angle labeled \( x^\circ \) is an exterior angle of the small triangle formed with angles \( 30^\circ \) and \( 40^\circ \). According to the exterior angle theorem, the exterior angle (here \( x \)) is equal to the sum of the two non-adjacent interior angles of the triangle. Thus, we have:

\[
x = 30^\circ + 40^\circ = 70^\circ
\]

Therefore, the value of \( x \) is:

\[
x = 70^\circ
\]

Would you like more details or have any further questions?

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Here are 5 related questions to explore further:
1. How does the exterior angle theorem apply to different types of triangles?
2. What are other methods to calculate exterior angles in geometric figures?
3. How can you calculate the sum of interior angles in any polygon?
4. How do right triangles help in calculating missing angles or sides in geometry?
5. What is the difference between an interior and exterior angle in geometry?

**Tip**: Always check for triangles where the sum of interior angles adds up to \(180^\circ\), as this helps in solving many geometry problems.

In the figure shown, what is the value of x?

This geometry problem asks for the value of an angle x, which can be solved using the Exterior Angle Theorem. The key steps involve understanding the properties of triangles and applying the theorem to find the exterior angle. Suitable for students in Grades 7-9.

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