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Calculate the size of angle w, x, y and z? 

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

### Step 1: Identify the angles
Let's label the angles based on the information provided in the image:

- \(w\): Angle at the top of the isosceles triangle (vertex angle).
- \(x\) and \(y\): Angles adjacent to each other, one of which forms a straight line with \(z\).
- \(z\): Angle that is external to the triangle.

### Step 2: Apply triangle and straight-line properties

1. **Angle \(w\)**:
   - Since the triangle is isosceles, the base angles of the triangle are equal. Let's denote each base angle as \( \alpha \).
   - In a triangle, the sum of the angles is \(180^\circ\).
   - Therefore, \(w + 2\alpha = 180^\circ\).

2. **Angle \(x\)**:
   - Angle \(x\) is adjacent to angle \(w\). In the triangle, \(x = \alpha\) since it's a base angle.

3. **Angle \(y\)**:
   - Angle \(y\) is external to the triangle. By the external angle theorem, \(y = w + x\).

4. **Angle \(z\)**:
   - Angle \(z\) forms a straight line with angle \(y\). Therefore, \(y + z = 180^\circ\).

### Step 3: Solve for each angle
Let's express each angle:

1. **Angle \(w\)**:  
   - If the sum of the angles in the triangle is \(180^\circ\), and the triangle is isosceles:
     \[
     w + 2x = 180^\circ
     \]

2. **Angle \(y\)**:  
   - Using the external angle theorem:
     \[
     y = w + x
     \]

3. **Angle \(z\)**:  
   - Since \(y\) and \(z\) are on a straight line:
     \[
     z = 180^\circ - y
     \]

### Conclusion
To calculate specific values, we would need more information, such as one of the angles. However, the general relationships between the angles are given above.

Would you like further clarification or have specific values to work with?

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**Related Questions:**
1. How can the external angle theorem be applied in different triangle types?
2. What are the properties of isosceles triangles?
3. How do you calculate the sum of angles in polygons?
4. What are the angle properties in parallel lines intersected by a transversal?
5. How do you determine angles in cyclic quadrilaterals?

**Tip:** Always remember that the sum of interior angles in any triangle is always \(180^\circ\).

Explore how to calculate the sizes of angles w, x, y, and z in a triangle using the sum of angles theorem and external angle theorem. This detailed solution covers properties of triangles and straight lines, suitable for high school students.

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