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To solve for \( x \) in the given triangle, let's analyze the diagram.

### Observations:
- The triangle \( ABC \) and \( BCD \) share side \( BC \).
- The angle \( \angle ABC \) and \( \angle BCD \) are given as equal, which suggests that triangles \( ABC \) and \( BCD \) might be similar.

### Approach:
1. **Identify the angles:**
   - \( \angle ABC = 8^\circ \) (labeled in red).
   - \( \angle BAC = 8^\circ \) (labeled in red).
   - \( \angle ABD = y \) (unknown, labeled in blue).

2. **Sum of angles in triangle \( ABC \):**
   \[
   \angle BAC + \angle ABC + \angle ACB = 180^\circ
   \]
   Substituting the known values:
   \[
   8^\circ + 8^\circ + \angle ACB = 180^\circ
   \]
   \[
   \angle ACB = 180^\circ - 16^\circ = 164^\circ
   \]

3. **Angle \( \angle DCB \) in triangle \( BCD \):**
   \[
   \angle DBC = \angle ACB = 164^\circ
   \]

4. **Applying the exterior angle theorem on \( \triangle ABC \):**
   \[
   x = \angle ACB
   \]
   Therefore, \( x = 164^\circ \).

So, the value of \( x \) is \( \boxed{164^\circ} \).

Let me know if you'd like further details or have any questions. Here are some related questions to consider:

1. What is the value of angle \( \angle ACB \) in triangle \( ABC \)?
2. Can you find the value of \( y \) in the diagram?
3. How does the sum of angles in a triangle help in solving for unknown angles?
4. What is the exterior angle theorem, and how is it applied in this context?
5. If triangles \( ABC \) and \( BCD \) are similar, what other properties can be derived?

**Tip:** When dealing with angles in triangles, always remember that the sum of the interior angles is 180 degrees. This fundamental property is key to solving many geometric problems.

Learn how to solve for angle x in triangles ABC and BCD using the angle sum property and the exterior angle theorem. Detailed step-by-step solution provided.

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