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To solve this problem, let's analyze the diagram and the given information:

1. **Angles at a Point:** The angles around a point sum up to \(360^\circ\).
2. **Given Equations:**
   - \(m\angle 5 = x\)
   - \(m\angle 9 = 2x + 40^\circ\)
   - \(m\angle 7 = 4x\)

Since angles \( \angle 5 \), \( \angle 6 \), \( \angle 7 \), \( \angle 8 \), \( \angle 9 \), and \( \angle 10 \) form a complete circle around the point, their measures sum to \(360^\circ\):

\[
m\angle 5 + m\angle 6 + m\angle 7 + m\angle 8 + m\angle 9 + m\angle 10 = 360^\circ
\]

### Step 1: Express the Angles
Using the given information, express all the angles:
- \(m\angle 5 = x\)
- \(m\angle 9 = 2x + 40^\circ\)
- \(m\angle 7 = 4x\)

Assume that \(m\angle 6 = y\), \(m\angle 8 = z\), and \(m\angle 10 = w\).

### Step 2: Utilize Linear Pairs
Since the angles on straight lines add up to \(180^\circ\), let's consider the pairs:
- \(m\angle 5 + m\angle 8 = 180^\circ\) \( \Rightarrow x + z = 180^\circ \) \( \Rightarrow z = 180^\circ - x\)
- \(m\angle 7 + m\angle 10 = 180^\circ\) \( \Rightarrow 4x + w = 180^\circ\) \( \Rightarrow w = 180^\circ - 4x\)
- \(m\angle 6 + m\angle 9 = 180^\circ\) \( \Rightarrow y + (2x + 40^\circ) = 180^\circ\) \( \Rightarrow y = 140^\circ - 2x\)

### Step 3: Sum of Angles
The sum of all angles equals \(360^\circ\):

\[
x + (140^\circ - 2x) + 4x + (180^\circ - x) + (2x + 40^\circ) + (180^\circ - 4x) = 360^\circ
\]

Simplifying:

\[
x + 140^\circ - 2x + 4x + 180^\circ - x + 2x + 40^\circ + 180^\circ - 4x = 360^\circ
\]

\[
360^\circ + x = 360^\circ \Rightarrow x = 0^\circ
\]

However, this is not possible because an angle cannot be \(0^\circ\). Let's instead sum the non-variable constants first and solve for \(x\):

### Step 4: Adjustments
Since \( x = 0^\circ \) is not possible, we can reexamine the equation assuming all angles contribute:

Given the total expression and constants, the calculation doesn't resolve neatly, suggesting an alternate approach or rechecking assumptions. If the math was consistent and led to zero, we'd conclude there's an error. Here, revisiting setup checks might help.

If we assume no error, the variable angles suggest \(x > 0^\circ\) hence contradiction in the problem, needing reassessment.

Would you like more details or to continue rechecking for accuracy? Here's how to approach similar problems:

1. **What if \(x\) were non-zero in an equation setup?**
2. **What geometric concepts validate angle equations?**
3. **Could you approach this with a substitution method?**
4. **What are alternative solutions in geometric contradictions?**
5. **How to approach similar setups in circle angles?**

**Tip:** Always ensure your angle measurements fit logical contexts, especially when adding results.

Explore how to solve angle problems using given equations. This detailed solution covers concepts like angles at a point, linear pairs, and circle properties. Suitable for students in Grades 7-9.

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