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The problem asks to find the value of \(x\) in a convex octagon, given the measures of its interior angles.

### Step 1: Formula for the sum of interior angles of an octagon
The sum \(S\) of the interior angles of any \(n\)-sided polygon can be calculated using the formula:
\[
S = 180(n - 2)
\]
For an octagon (\(n = 8\)):
\[
S = 180(8 - 2) = 180(6) = 1080^\circ
\]
So, the sum of the interior angles of the octagon is \(1080^\circ\).

### Step 2: Setting up the equation
We are given eight interior angles:

1. \(x + 55^\circ\)
2. \(3x + 20^\circ\)
3. \(4x^\circ\)
4. \(4x - 10^\circ\)
5. \(6x - 55^\circ\)
6. \(3x + 52^\circ\)
7. \(3x^\circ\)
8. \(2x + 30^\circ\)

Summing these expressions, the total must equal \(1080^\circ\). So, we set up the equation:
\[
(x + 55) + (3x + 20) + 4x + (4x - 10) + (6x - 55) + (3x + 52) + 3x + (2x + 30) = 1080
\]

### Step 3: Simplifying the equation
Combine like terms:
\[
x + 3x + 4x + 4x + 6x + 3x + 3x + 2x = 26x
\]
And the constants:
\[
55 + 20 - 10 - 55 + 52 + 30 = 92
\]
Now, the equation becomes:
\[
26x + 92 = 1080
\]

### Step 4: Solving for \(x\)
Subtract 92 from both sides:
\[
26x = 1080 - 92
\]
\[
26x = 988
\]
Divide both sides by 26:
\[
x = \frac{988}{26} = 38
\]

### Final Answer:
The value of \(x\) is \(38\).

Would you like a further breakdown or clarification on any part of this solution?

Here are five related questions for additional practice:
1. What would be the value of \(x\) if this were a hexagon with the same angle expressions?
2. How would the sum of interior angles change for a decagon?
3. What is the smallest angle in the octagon when \(x = 38\)?
4. How does the formula for the sum of interior angles apply to a quadrilateral?
5. Can the interior angles of a polygon ever sum to less than 180°?

**Tip**: Always remember to double-check your work by adding up the individual angles to ensure they sum to the correct total for the polygon.

A convex octagon has interior angles with measures (x + 55)°, (3x + 20)°, 4x°, (4x - 10)°, (6x - 55)°, (3x + 52)°, 3x°, and (2x + 30)°. Find the value of x.

Learn how to solve for x in a convex octagon with interior angles expressed as (x + 55)°, (3x + 20)°, 4x°, (4x - 10)°, (6x - 55)°, (3x + 52)°, 3x°, and (2x + 30)°. The solution involves using the sum of interior angles formula for an octagon, algebraic equations, and combining like terms. Suitable for high school students in Grades 9-12.

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