One interior angle of an irregular polygon is 800 and each of the other interior angles is 1250.find the number of the sides of the polygon .

To solve this problem, we need to use the properties of polygons, specifically the sum of interior angles.

### Step 1: Determine the sum of the interior angles
The formula to calculate the sum of the interior angles of a polygon with \(n\) sides is:
\[
\text{Sum of interior angles} = 180^\circ \times (n - 2)
\]
where \(n\) is the number of sides of the polygon.

### Step 2: Analyze the given angles
We know the following:
- One interior angle is \(800^\circ\).
- All other interior angles are \(1250^\circ\).

Let the polygon have \(n\) sides. Out of these \(n\) sides:
- One angle is \(800^\circ\).
- The remaining \(n-1\) angles are \(1250^\circ\).

Thus, the sum of the interior angles is:
\[
800^\circ + 1250^\circ \times (n - 1)
\]

### Step 3: Set up the equation for the sum of the angles
We can equate the two expressions for the sum of the interior angles:
\[
800^\circ + 1250^\circ \times (n - 1) = 180^\circ \times (n - 2)
\]

### Step 4: Solve for \(n\)
Now, solve the equation:
\[
800 + 1250(n - 1) = 180(n - 2)
\]
Distribute the terms:
\[
800 + 1250n - 1250 = 180n - 360
\]
Simplify:
\[
1250n - 450 = 180n - 360
\]
Move all terms involving \(n\) to one side and constant terms to the other side:
\[
1250n - 180n = -360 + 450
\]
Simplify:
\[
1070n = 90
\]
Solve for \(n\):
\[
n = \frac{90}{1070} =  \ text

Learn how to find the number of sides of an irregular polygon when one interior angle is 800° and all others are 1250°. Detailed step-by-step solution using the formula for the sum of interior angles of a polygon. Suitable for high school students studying geometry and polygons.

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