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To solve for \(x + y + z + w\) in the given figure, we can use the properties of angles, specifically focusing on straight lines and triangles.

### Step-by-Step Solution:

1. **Identify the angles:**
   - The angle at point \(B\) is given as \(120^\circ\).
   - The angle at point \(C\) is given as \(60^\circ\).
   - \( \angle ABC = y \) (adjacent to \(120^\circ\)).
   - \( \angle BAC = z \).
   - \( \angle DAB = w \).
   - \( \angle BCA = x \).

2. **Use the property of straight lines:**
   - On a straight line, the sum of adjacent angles is \(180^\circ\).
   - At point \(B\), we have \(y + 120^\circ = 180^\circ\).
     \[
     y = 180^\circ - 120^\circ = 60^\circ
     \]
   - At point \(C\), we have \(x + 60^\circ = 180^\circ\).
     \[
     x = 180^\circ - 60^\circ = 120^\circ
     \]

3. **Use the property of triangles:**
   - The sum of the angles in a triangle is \(180^\circ\).
   - For triangle \(ABC\):
     \[
     z + 60^\circ + 120^\circ = 180^\circ
     \]
     \[
     z + 180^\circ = 180^\circ \Rightarrow z = 0^\circ
     \]

   - For triangle \(ABD\):
     \[
     w + z + 60^\circ = 180^\circ
     \]
     Since \(z = 0^\circ\):
     \[
     w + 60^\circ = 180^\circ
     \]
     \[
     w = 180^\circ - 60^\circ = 120^\circ
     \]

4. **Calculate \(x + y + z + w\):**
   \[
   x + y + z + w = 120^\circ + 60^\circ + 0^\circ + 120^\circ = 300^\circ
   \]

### Final Answer:
\[
\boxed{300^\circ}
\]

Would you like more details or have any other questions? 

### Follow-up Questions:
1. How would the problem change if one of the angles was not given?
2. Can you explain why the sum of angles on a straight line is \(180^\circ\)?
3. What are the properties of exterior angles in a triangle?
4. How can you use the properties of parallel lines to solve angle problems?
5. What happens if you have a quadrilateral instead of a triangle?
6. How do you solve for an unknown angle in a right triangle?
7. What is the significance of supplementary angles?
8. How do you apply the concept of congruent triangles to solve geometry problems?

### Tip:
Always double-check the relationships between angles in diagrams, especially focusing on lines that are straight or parallel, as they often provide the most useful information.

Learn how to solve for x + y + z + w in a geometric figure using properties of angles, triangles, and straight lines. This problem involves calculating angles and applying mathematical theorems. Suitable for high school students studying geometry.

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