To determine the value of [tex]\( n \)[/tex], the number of sides of the polygon, let's follow a step-by-step approach.
1. Recall the formula for the sum of the interior angles of an [tex]\( n \)[/tex]-sided polygon:
[tex]\[ \text{Sum of interior angles} = 180(n-2) \][/tex]
This formula tells us that the sum of the interior angles of an [tex]\( n \)[/tex]-sided polygon is [tex]\( 180(n-2) \)[/tex] degrees.
2. Identify the known interior angles:
- Two of the interior angles are given: 74° and 136°.
- The remaining interior angles are each 110°.
3. Let [tex]\( n \)[/tex] be the number of sides of the polygon. Since we already have 2 specific angles (74° and 136°), the remaining [tex]\( (n - 2) \)[/tex] angles are each 110°.
Therefore, the sum of the remaining interior angles can be written as:
[tex]\[ 110(n - 2) \][/tex]
4. Express the total sum of all interior angles using the given angles and the remaining angles:
[tex]\[ 74 + 136 + 110(n - 2) \][/tex]
5. Set up the equation using the sum of the interior angles formula:
[tex]\[
74 + 136 + 110(n - 2) = 180(n - 2)
\][/tex]
6. Combine like terms and simplify the equation:
[tex]\[
210 + 110(n - 2) = 180(n - 2)
\][/tex]
Simplify further:
[tex]\[
210 + 110n - 220 = 180n - 360
\][/tex]
[tex]\[
-10 + 110n = 180n - 360
\][/tex]
7. Isolate the variable [tex]\( n \)[/tex]:
First, add 360 to both sides to remove -360 from the right side:
[tex]\[
350 + 110n = 180n
\][/tex]
Then, subtract 110n from both sides to isolate [tex]\( n \)[/tex]:
[tex]\[
350 = 70n
\][/tex]
8. Solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{350}{70}
\][/tex]
[tex]\[
n = 5
\][/tex]
Thus, the polygon has 5 sides. So, [tex]\( n = 5 \)[/tex].
