Question

practice
polygons and angles
find the sum of the interior angle measures of each polygon.

1. pentagon
2. decagon
3. 16 - gon
4. 18 - gon
5. 30 - gon
6. 34 - gon

find the measure of one interior angle in each regular polygon.
round to the nearest tenth if necessary.

1. pentagon
2. octagon
3. 24 - gon

algebra for exercises 10 and 11, determine the angle measures in each polygon.

1. diagram of a parallelogram with angles (x^circ), (5x^circ), (x^circ), (5x^circ)
2. diagram of a pentagon with angles (135^circ), (135^circ), (x^circ), (x^circ), (x^circ)
3. flooring a floor is tiled with a pattern consisting of regular octagons and squares as shown. find the measure of each angle at the circled vertex. then find the sum of the angles.
4. art rachaunn is laying out a pattern for a stained glass window. so far he has placed the 13 regular polygons shown. find the measure of each angle at the circled vertex. then find the sum of the angles.
5. reasoning vanessa’s mother made a quilt using a pattern of repeating regular hexagons as shown. will vanessa be able to make a similar quilt with a pattern of repeating regular pentagons? explain your reasoning.

3 • chapter 5 triangles and the pythagorean theorem

Explanation:

Step1: Sum of interior angles formula

The formula for the sum of interior angles of an $n$-sided polygon is $(n-2) \times 180^\circ$.

Step2: Solve 1-6 (sum of interior angles)

1. Pentagon ($n=5$)

$$\begin{align*} (5-2) \times 180^\circ &= 3 \times 180^\circ = 540^\circ \end{align*}$$

2. Decagon ($n=10$)

$$\begin{align*} (10-2) \times 180^\circ &= 8 \times 180^\circ = 1440^\circ \end{align*}$$

3. 16-gon ($n=16$)

$$\begin{align*} (16-2) \times 180^\circ &= 14 \times 180^\circ = 2520^\circ \end{align*}$$

4. 18-gon ($n=18$)

$$\begin{align*} (18-2) \times 180^\circ &= 16 \times 180^\circ = 2880^\circ \end{align*}$$

5. 30-gon ($n=30$)

$$\begin{align*} (30-2) \times 180^\circ &= 28 \times 180^\circ = 5040^\circ \end{align*}$$

6. 34-gon ($n=34$)

$$\begin{align*} (34-2) \times 180^\circ &= 32 \times 180^\circ = 5760^\circ \end{align*}$$

Step3: Solve 7-9 (one interior angle, regular polygon)

Use $\frac{(n-2) \times 180^\circ}{n}$ for one interior angle.

7. Regular pentagon ($n=5$)

$$\begin{align*} \frac{(5-2) \times 180^\circ}{5} &= \frac{540^\circ}{5} = 108^\circ \end{align*}$$

8. Regular octagon ($n=8$)

$$\begin{align*} \frac{(8-2) \times 180^\circ}{8} &= \frac{1080^\circ}{8} = 135^\circ \end{align*}$$

9. Regular 24-gon ($n=24$)

$$\begin{align*} \frac{(24-2) \times 180^\circ}{24} &= \frac{3960^\circ}{24} = 165^\circ \end{align*}$$

Step4: Solve 10 (parallelogram angle sum)

Parallelogram has $n=4$, sum of angles is $360^\circ$. Set up equation:

$$\begin{align*} x + 5x + x + 5x &= 360^\circ \\ 12x &= 360^\circ \\ x &= 30^\circ \end{align*}$$

Angles: $30^\circ, 150^\circ, 30^\circ, 150^\circ$

Step5: Solve 11 (pentagon angle sum)

Pentagon sum is $540^\circ$. Set up equation:

$$\begin{align*} x + x + 135^\circ + 135^\circ + x &= 540^\circ \\ 3x + 270^\circ &= 540^\circ \\ 3x &= 270^\circ \\ x &= 90^\circ \end{align*}$$

Angles: $90^\circ, 90^\circ, 135^\circ, 135^\circ, 90^\circ$

Step6: Solve 12 (floor tiling angles)

Regular octagon interior angle is $135^\circ$, square interior angle is $90^\circ$.
At the circled vertex: octagon angle $=135^\circ$, square angle $=90^\circ$.
Sum: $135^\circ + 90^\circ + 135^\circ = 360^\circ$

Step7: Solve 13 (stained glass angles)

13-gon interior angle:

$$\begin{align*} \frac{(13-2) \times 180^\circ}{13} &= \frac{1980^\circ}{13} \approx 152.3^\circ \end{align*}$$

Sum at circled vertex (3 polygons): $152.3^\circ \times 3 \approx 456.9^\circ$

Step8: Solve 14 (tiling with pentagons)

Regular pentagon interior angle is $108^\circ$. For tiling, the sum of angles at a vertex must be $360^\circ$.
$360^\circ \div 108^\circ = \frac{10}{3} \approx 3.33$, which is not an integer. So pentagons cannot tile without gaps.

Answer:

1. $540^\circ$
2. $1440^\circ$
3. $2520^\circ$
4. $2880^\circ$
5. $5040^\circ$
6. $5760^\circ$
7. $108^\circ$
8. $135^\circ$
9. $165^\circ$
10. $30^\circ, 150^\circ, 30^\circ, 150^\circ$
11. $90^\circ, 90^\circ, 135^\circ, 135^\circ, 90^\circ$
12. Octagon angle: $135^\circ$, square angle: $90^\circ$; Sum: $360^\circ$
13. One 13-gon angle: $\approx 152.3^\circ$; Sum: $\approx 456.9^\circ$
14. No, regular pentagons cannot tile a plane. Their interior angle is $108^\circ$, and $360^\circ$ is not evenly divisible by $108^\circ$, so gaps would form at the vertices.