To calculate the number of sides [tex]\( n \)[/tex] of the polygon, we need to follow these steps:
1. Determine the sum of the known angles:
The sum of three known interior angles is 510 degrees.
2. Calculate the number of the other angles:
Since each of the other interior angles measures 128 degrees, we need to determine how many such angles there are. This can be found by dividing the sum of the known angles by the measurement of one of the other interior angles.
[tex]\[
\text{Number of other angles} = \frac{510}{128} \approx 3.984375
\][/tex]
3. Determine the total sum of the interior angles:
The total sum of the interior angles can be found by adding the sum of the known angles and the contribution from each of the other interior angles.
[tex]\[
\text{Total sum of interior angles} = 510 + 3.984375 \times 128 = 1020
\][/tex]
4. Use the formula for the sum of interior angles:
The sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by the formula:
[tex]\[
(n - 2) \times 180
\][/tex]
Set this equal to the total sum of the interior angles and solve for [tex]\( n \)[/tex]:
[tex]\[
(n - 2) \times 180 = 1020
\][/tex]
[tex]\[
n - 2 = \frac{1020}{180} = 5.66666
\][/tex]
Add 2 to both sides to find [tex]\( n \)[/tex]:
[tex]\[
n = 5.66666 + 2 = 7.66666
\][/tex]
So, the polygon has approximately 7.67 sides. Since the number of sides [tex]\( n \)[/tex] in a polygon must be an integer, this problem might have involved some rounding or may have different constraints or interpretations. If we interpret it as needing a whole number, adjustments or conditions of rounding may need to be considered.
