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To solve for the angle \( x \) in the given triangle:

### Step-by-Step Solution:
1. **Identify Angles in the Triangle:**
   - The angle \( \angle BAC \) is labeled as \( 2x \).
   - The angle \( \angle BCA \) is labeled as \( x \).
   - The exterior angle \( \angle DAB \) outside the triangle at point \( A \) is labeled as \( 3x \).

2. **Use the Exterior Angle Theorem:**
   The exterior angle of a triangle is equal to the sum of the two opposite interior angles. Thus, we have:
   \[
   3x = 2x + x
   \]

3. **Solve for \( x \):**
   - Combine like terms on the right side:
     \[
     3x = 3x
     \]
   - This equation confirms that the relationship holds for all \( x \). Therefore, this configuration is consistent with any positive value of \( x \) that satisfies the angle sum property of a triangle.
   
4. **Using the Triangle Sum Property:**
   In a triangle, the sum of interior angles must be \( 180^\circ \).
   \[
   2x + x + x = 180^\circ
   \]
   Simplify this:
   \[
   4x = 180^\circ
   \]
   \[
   x = 45^\circ
   \]

### Conclusion:
The angle \( x = 45^\circ \).

Would you like a further breakdown, or do you have any questions?

Here are a few additional questions related to this problem:
1. How would the solution change if the exterior angle were labeled differently?
2. What is the significance of the exterior angle theorem in triangle geometry?
3. How can we verify the sum of angles in any triangle is always \( 180^\circ \)?
4. How would this solution be affected if the triangle were not a simple triangle?
5. Can you find other properties of this triangle based on the angles given?

**Tip:** Always remember that the exterior angle theorem is a quick way to solve for unknown angles in many triangle problems!

Find the Angle x in the given triangle with angles 3x, 2x, and x.

Learn how to solve for the angle x in a triangle using the exterior angle theorem and the angle sum property. This problem involves basic triangle geometry and is suitable for students in Grades 8-10.

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