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To solve for \( z \) in this problem, we can use the fact that the angles around a point sum to \( 360^\circ \). The three angles provided are \( 39^\circ \), \( z + 29^\circ \), and \( 4z + 36^\circ \).

### Step-by-Step Solution

1. **Set up the equation:**
   Since the three angles must add up to \( 360^\circ \), we can write the equation as:
   \[
   39 + (z + 29) + (4z + 36) = 360
   \]

2. **Simplify the equation:**
   Combine like terms:
   \[
   39 + z + 29 + 4z + 36 = 360
   \]
   \[
   5z + 104 = 360
   \]

3. **Solve for \( z \):**
   Subtract \( 104 \) from both sides:
   \[
   5z = 256
   \]
   Divide by \( 5 \):
   \[
   z = 51.2
   \]

### Answer
The value of \( z \) is \( 51.2^\circ \).

Would you like further details or explanations on any of the steps?

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**Related Questions:**

1. How do you find the value of an unknown angle in a triangle?
2. How can angle relationships help in solving for unknown variables?
3. What are supplementary and complementary angles?
4. How can we use algebra to solve problems involving angles?
5. What are the steps to solve an equation with multiple terms and variables?

**Tip:** Remember, the sum of angles around a point is always \( 360^\circ \). This fact is useful in many geometry problems.

What is the value of z? The angles around a point sum to 360°, with the angles 39°, z + 29°, and 4z + 36°.

Learn how to find the value of z using the angle sum property, where three angles around a point add up to 360°. This solution involves setting up an equation and solving for the unknown variable z. Suitable for Grades 8-10.

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