Question

The sum of the measures of the interior angles of a polygon with$n$ sides is $S$. Without using$n$ in your answer, express in terms of$S$ the sum of the measures of the angles of a polygon with:

1. $\mathit{n}\mathbf{+}\mathbf{1}$sides
2. $\mathbf{2}\mathit{n}$sides

Short answer

1. The sum of the measures of the angles of a polygon with$n+1$ sides is$\mathit{S}\mathbf{+}\mathbf{180}$
2. The sum of the measures of the angles of a polygon with$2n$ sides is$\mathbf{2}\mathit{S}\mathbf{+}\mathbf{360}$

Step-by-step solution

Part a. Step 1. Write the theorem for the sum of interior angles for a polygon.

The theorem for the sum of interior angles for a polygon states that the sum of the measures of the interior angles for a polygon with sides is $\left(n-2\right)\mathbf{180}$.

Part a. Step 2. Write the relation between S and n.

The sum$S$ of the measures of the interior angles for a polygon with$n$ sides is given by:

$S=\left(n-2\right)180$

Therefore, the relation between$S$ and$n$ is$S=\left(n-2\right)180$

Part a. Step 3. Write S in terms of n.

The value of$n$ in terms of$S$ is given by:

$$\begin{gathered} S=\left(n-2\right)180 \\ \frac{S}{180}=n-2 \\ \frac{S}{180}+2=n \\ \frac{S+360}{180}=n \end{gathered}$$

Part a. Step 4. Find the sum of the measures of the interior angle of the polygon with n+1 sides.

The sum of the measures of the interior angles for a polygon with$n+1$ sides is given by:

$\left(n+1-2\right)180=\left(n-1\right)180$

Part a. Step 5. Find the sum of the measures of the interior angle of the polygon with n+1 sides in terms of S.

The sum of the measures of the interior angles for a polygon with$n+1$ sides in terms of$S$ is given by:

$$\begin{gathered} \left(n-1\right)180=\left(\frac{S+360}{180}-1\right)180 \\ =\left(\frac{S+360-180}{180}\right)180 \\ =\left(\frac{S+180}{180}\right)180 \\ =S+180 \end{gathered}$$

Part b. Step 1. Write the theorem for the sum of interior angles for a polygon.

The theorem for the sum of interior angles for a polygon states that the sum of the measures of the interior angles for a polygon with sides is $\left(n-2\right)\mathbf{180}$.

Part b. Step 2. Write the relation between S and n.

The sum$S$ of the measures of the interior angles for a polygon with$n$ sides is given by:

$S=\left(n-2\right)180$

Therefore, the relation between$S$ and$n$ is$S=\left(n-2\right)180$

Part b. Step 3. Write S in terms of n.

The value of$n$ in terms of$S$ is given by:

$$\begin{gathered} S=\left(n-2\right)180 \\ \frac{S}{180}=n-2 \\ \frac{S}{180}+2=n \\ \frac{S+360}{180}=n \end{gathered}$$

Part b. Step 4. Find the sum of the measures of the interior angle of the polygon with 2n sides.

The sum of the measures of the interior angles for a polygon with$2n$ sides is given by:

$\left(2n-2\right)180=2\left(n-1\right)180$

Part b. Step 5. Find the sum of the measures of the interior angle of the polygon with 2n sides in terms of S.

The sum of the measures of the interior angles for a polygon with$2n$ sides in terms of$S$ is given by:

$$\begin{gathered} 2\left(n-1\right)180=2\left(\frac{S+360}{180}-1\right)180 \\ =2\left(\frac{S+360-180}{180}\right)180 \\ =2\left(\frac{S+180}{180}\right)180 \\ =2\left(S+180\right) \\ =2S+360 \end{gathered}$$