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In questions 4-6, use the regular polygon to solve the following problems:

4. What is the sum of the measures of the interior angles of the polygon?

[tex]\[
\begin{align*}
\text{Sum of interior angles} &= (n-2) \times 180^\circ \\
\text{For } n = 12, \text{ sum} &= (12-2) \times 180^\circ = 1800^\circ
\end{align*}
\][/tex]

5. Find the value of [tex]x[/tex].

[tex]\[
\frac{12}{1800} = \frac{150}{23x-11}
\][/tex]

6. What is the sum of the interior angles of a regular polygon with the given number of sides?

A. 100
B. 30
C. 50
D. 1,000

Answer :

Sure! Let's tackle this step by step.

### Part 4: Sum of the Interior Angles of the Polygon

To find the sum of the interior angles of a regular polygon, we use the formula:

[tex]\[
(n - 2) \times 180
\][/tex]

where [tex]\( n \)[/tex] is the number of sides of the polygon. We need the sum to match 1800. So, let's solve for [tex]\( n \)[/tex]:

[tex]\[
(n - 2) \times 180 = 1800
\][/tex]

Simplify it to find [tex]\( n \)[/tex]:

[tex]\[
n - 2 = \frac{1800}{180} = 10
\][/tex]
[tex]\[
n = 10 + 2 = 12
\][/tex]

So, the polygon has 12 sides.

### Part 5: Finding the Value of [tex]\( x \)[/tex]

For this part, we're given the equation:

[tex]\[
\frac{12}{1800} = \frac{150}{23x - 11}
\][/tex]

To solve for [tex]\( x \)[/tex], we can cross-multiply:

[tex]\[
12 \times (23x - 11) = 150 \times 1800
\][/tex]

Which simplifies to:

[tex]\[
12 \times 23x - 12 \times 11 = 270000
\][/tex]

Solving for [tex]\( x \)[/tex], we get:

[tex]\[
276x - 132 = 270000
\][/tex]
[tex]\[
276x = 270000 + 132
\][/tex]
[tex]\[
276x = 270132
\][/tex]
[tex]\[
x = \frac{270132}{276}
\][/tex]

This gives us [tex]\( x = \frac{22511}{23} \)[/tex], which is approximately 978.304.

### Part 6: Sum of the Interior Angles for 30 Sides

For a polygon with 30 sides, use the formula:

[tex]\[
(n - 2) \times 180
\][/tex]

Substituting [tex]\( n = 30 \)[/tex]:

[tex]\[
(30 - 2) \times 180 = 28 \times 180 = 5040
\][/tex]

Thus, the sum of the interior angles for a polygon with 30 sides is 5040 degrees.

I hope this helps clarify the solution! Let me know if you have any more questions.

Thanks for taking the time to read In questions 4 6 use the regular polygon to solve the following problems 4 What is the sum of the measures of the interior angles. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

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