Please answer, need to pass ASAP. I still have many questions to ... - Free Printable

To solve the problems in the worksheet, we need to use the following key formulas related to polygons:

1. Interior Angle Sum Formula:
\[
\text{Interior Angle Sum} = 180^\circ \times (n - 2)
\]
where \( n \) is the number of sides of the polygon.

2. Measure of One Interior Angle in a Regular Polygon:
\[
\text{Measure of ONE Interior Angle} = \frac{\text{Interior Angle Sum}}{n}
\]

3. Exterior Angle Sum Formula:
\[
\text{Exterior Angle Sum} = 360^\circ
\]
This is constant for any polygon.

4. Measure of One Exterior Angle in a Regular Polygon:
\[
\text{Measure of ONE Exterior Angle} = \frac{360^\circ}{n}
\]

Now, let's solve each part of the table step by step.

---

Row 1: General Case (\( n \) sides)

- # Sides: \( n \)
- Interior Angle Sum:
\[
180^\circ \times (n - 2)
\]
- Measure of ONE Interior Angle:
\[
\frac{180^\circ \times (n - 2)}{n}
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{n}
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{# Sides} & \text{Interior Angle Sum} & \text{Measure of ONE Interior Angle} & \text{Exterior Angle Sum} & \text{Measure of ONE Exterior Angle} \\
\hline
n & 180^\circ \times (n - 2) & \frac{180^\circ \times (n - 2)}{n} & 360^\circ & \frac{360^\circ}{n} \\
\hline
\end{array}
\]

---

Row 2: \( n = 14 \)

- # Sides: 14
- Interior Angle Sum:
\[
180^\circ \times (14 - 2) = 180^\circ \times 12 = 2160^\circ
\]
- Measure of ONE Interior Angle:
\[
\frac{2160^\circ}{14} \approx 154.3^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{14} \approx 25.7^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
14 & 2160^\circ & 154.3^\circ & 360^\circ & 25.7^\circ \\
\hline
\end{array}
\]

---

Row 3: \( n = 24 \)

- # Sides: 24
- Interior Angle Sum:
\[
180^\circ \times (24 - 2) = 180^\circ \times 22 = 3960^\circ
\]
- Measure of ONE Interior Angle:
\[
\frac{3960^\circ}{24} = 165^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{24} = 15^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
24 & 3960^\circ & 165^\circ & 360^\circ & 15^\circ \\
\hline
\end{array}
\]

---

Row 4: \( n = 17 \)

- # Sides: 17
- Interior Angle Sum:
\[
180^\circ \times (17 - 2) = 180^\circ \times 15 = 2700^\circ
\]
- Measure of ONE Interior Angle:
\[
\frac{2700^\circ}{17} \approx 158.8^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{17} \approx 21.2^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
17 & 2700^\circ & 158.8^\circ & 360^\circ & 21.2^\circ \\
\hline
\end{array}
\]

---

Row 5: Interior Angle Sum = 1080°

- Interior Angle Sum: 1080°
- # Sides:
\[
180^\circ \times (n - 2) = 1080^\circ \implies n - 2 = \frac{1080^\circ}{180^\circ} = 6 \implies n = 8
\]
- Measure of ONE Interior Angle:
\[
\frac{1080^\circ}{8} = 135^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{8} = 45^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
8 & 1080^\circ & 135^\circ & 360^\circ & 45^\circ \\
\hline
\end{array}
\]

---

Row 6: Interior Angle Sum = 900°

- Interior Angle Sum: 900°
- # Sides:
\[
180^\circ \times (n - 2) = 900^\circ \implies n - 2 = \frac{900^\circ}{180^\circ} = 5 \implies n = 7
\]
- Measure of ONE Interior Angle:
\[
\frac{900^\circ}{7} \approx 128.6^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{7} \approx 51.4^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
7 & 900^\circ & 128.6^\circ & 360^\circ & 51.4^\circ \\
\hline
\end{array}
\]

---

Row 7: Interior Angle Sum = 5040°

- Interior Angle Sum: 5040°
- # Sides:
\[
180^\circ \times (n - 2) = 5040^\circ \implies n - 2 = \frac{5040^\circ}{180^\circ} = 28 \implies n = 30
\]
- Measure of ONE Interior Angle:
\[
\frac{5040^\circ}{30} = 168^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{30} = 12^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
30 & 5040^\circ & 168^\circ & 360^\circ & 12^\circ \\
\hline
\end{array}
\]

---

Row 8: Interior Angle Sum = 1620°

- Interior Angle Sum: 1620°
- # Sides:
\[
180^\circ \times (n - 2) = 1620^\circ \implies n - 2 = \frac{1620^\circ}{180^\circ} = 9 \implies n = 11
\]
- Measure of ONE Interior Angle:
\[
\frac{1620^\circ}{11} \approx 147.3^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{11} \approx 32.7^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
11 & 1620^\circ & 147.3^\circ & 360^\circ & 32.7^\circ \\
\hline
\end{array}
\]

---

Row 9: Measure of ONE Interior Angle = 150°

- Measure of ONE Interior Angle: 150°
- # Sides:
\[
\text{Measure of ONE Interior Angle} = \frac{180^\circ \times (n - 2)}{n} = 150^\circ
\]
\[
180^\circ \times (n - 2) = 150^\circ \times n \implies 180n - 360 = 150n \implies 30n = 360 \implies n = 12
\]
- Interior Angle Sum:
\[
180^\circ \times (12 - 2) = 180^\circ \times 10 = 1800^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{12} = 30^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
12 & 1800^\circ & 150^\circ & 360^\circ & 30^\circ \\
\hline
\end{array}
\]

---

Row 10: Measure of ONE Exterior Angle = 120°

- Measure of ONE Exterior Angle: 120°
- # Sides:
\[
\text{Measure of ONE Exterior Angle} = \frac{360^\circ}{n} = 120^\circ \implies n = \frac{360^\circ}{120^\circ} = 3
\]
- Interior Angle Sum:
\[
180^\circ \times (3 - 2) = 180^\circ \times 1 = 180^\circ
\]
- Measure of ONE Interior Angle:
\[
\frac{180^\circ}{3} = 60^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
3 & 180^\circ & 60^\circ & 360^\circ & 120^\circ \\
\hline
\end{array}
\]

---

Row 11: Measure of ONE Interior Angle = 156°

- Measure of ONE Interior Angle: 156°
- # Sides:
\[
\text{Measure of ONE Interior Angle} = \frac{180^\circ \times (n - 2)}{n} = 156^\circ
\]
\[
180^\circ \times (n - 2) = 156^\circ \times n \implies 180n - 360 = 156n \implies 24n = 360 \implies n = 15
\]
- Interior Angle Sum:
\[
180^\circ \times (15 - 2) = 180^\circ \times 13 = 2340^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]
- Measure of ONE Exterior Angle:
\[
\frac{360^\circ}{15} = 24^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
15 & 2340^\circ & 156^\circ & 360^\circ & 24^\circ \\
\hline
\end{array}
\]

---

Row 12: Measure of ONE Exterior Angle = 10°

- Measure of ONE Exterior Angle: 10°
- # Sides:
\[
\text{Measure of ONE Exterior Angle} = \frac{360^\circ}{n} = 10^\circ \implies n = \frac{360^\circ}{10^\circ} = 36
\]
- Interior Angle Sum:
\[
180^\circ \times (36 - 2) = 180^\circ \times 34 = 6120^\circ
\]
- Measure of ONE Interior Angle:
\[
\frac{6120^\circ}{36} = 170^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
36 & 6120^\circ & 170^\circ & 360^\circ & 10^\circ \\
\hline
\end{array}
\]

---

Row 13: Measure of ONE Exterior Angle = 7.2°

- Measure of ONE Exterior Angle: 7.2°
- # Sides:
\[
\text{Measure of ONE Exterior Angle} = \frac{360^\circ}{n} = 7.2^\circ \implies n = \frac{360^\circ}{7.2^\circ} = 50
\]
- Interior Angle Sum:
\[
180^\circ \times (50 - 2) = 180^\circ \times 48 = 8640^\circ
\]
- Measure of ONE Interior Angle:
\[
\frac{8640^\circ}{50} = 172.8^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
50 & 8640^\circ & 172.8^\circ & 360^\circ & 7.2^\circ \\
\hline
\end{array}
\]

---

Row 14: Measure of ONE Exterior Angle = 90°

- Measure of ONE Exterior Angle: 90°
- # Sides:
\[
\text{Measure of ONE Exterior Angle} = \frac{360^\circ}{n} = 90^\circ \implies n = \frac{360^\circ}{90^\circ} = 4
\]
- Interior Angle Sum:
\[
180^\circ \times (4 - 2) = 180^\circ \times 2 = 360^\circ
\]
- Measure of ONE Interior Angle:
\[
\frac{360^\circ}{4} = 90^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
4 & 360^\circ & 90^\circ & 360^\circ & 90^\circ \\
\hline
\end{array}
\]

---

Row 15: Measure of ONE Exterior Angle = 5°

- Measure of ONE Exterior Angle: 5°
- # Sides:
\[
\text{Measure of ONE Exterior Angle} = \frac{360^\circ}{n} = 5^\circ \implies n = \frac{360^\circ}{5^\circ} = 72
\]
- Interior Angle Sum:
\[
180^\circ \times (72 - 2) = 180^\circ \times 70 = 12600^\circ
\]
- Measure of ONE Interior Angle:
\[
\frac{12600^\circ}{72} = 175^\circ
\]
- Exterior Angle Sum:
\[
360^\circ
\]

So, the row is:
\[
\begin{array}{|c|c|c|c|c|}
\hline
72 & 12600^\circ & 175^\circ & 360^\circ & 5^\circ \\
\hline
\end{array}
\]

---

Final Answer

\[
\boxed{
\begin{array}{|c|c|c|c|c|}
\hline
\text{# Sides} & \text{Interior Angle Sum} & \text{Measure of ONE Interior Angle} & \text{Exterior Angle Sum} & \text{Measure of ONE Exterior Angle} \\
\hline
n & 180^\circ \times (n - 2) & \frac{180^\circ \times (n - 2)}{n} & 360^\circ & \frac{360^\circ}{n} \\
\hline
14 & 2160^\circ & 154.3^\circ & 360^\circ & 25.7^\circ \\
\hline
24 & 3960^\circ & 165^\circ & 360^\circ & 15^\circ \\
\hline
17 & 2700^\circ & 158.8^\circ & 360^\circ & 21.2^\circ \\
\hline
8 & 1080^\circ & 135^\circ & 360^\circ & 45^\circ \\
\hline
7 & 900^\circ & 128.6^\circ & 360^\circ & 51.4^\circ \\
\hline
30 & 5040^\circ & 168^\circ & 360^\circ & 12^\circ \\
\hline
11 & 1620^\circ & 147.3^\circ & 360^\circ & 32.7^\circ \\
\hline
12 & 1800^\circ & 150^\circ & 360^\circ & 30^\circ \\
\hline
3 & 180^\circ & 60^\circ & 360^\circ & 120^\circ \\
\hline
15 & 2340^\circ & 156^\circ & 360^\circ & 24^\circ \\
\hline
36 & 6120^\circ & 170^\circ & 360^\circ & 10^\circ \\
\hline
50 & 8640^\circ & 172.8^\circ & 360^\circ & 7.2^\circ \\
\hline
4 & 360^\circ & 90^\circ & 360^\circ & 90^\circ \\
\hline
72 & 12600^\circ & 175^\circ & 360^\circ & 5^\circ \\
\hline
\end{array}
}
\]