find the measure of one interior angle in each regular polygon. round your answer to the nearest tenth if necessary.
14) regular decagon
15) regular 22 - gon
16) regular 16 - gon
17) regular 15 - gon
find the measure of one exterior angle in each regular polygon. round your answer to the nearest tenth if necessary.
18) regular heptagon
19) regular 13 - gon
20) regular quadrilateral
21) regular hexagon

Question

Accepted Answer

### 14) Regular Decagon (Interior Angle)
# Explanation:
## Step1: Recall the formula for the sum of interior angles of a polygon.
The sum of interior angles of a polygon with $ n $ sides is $ S=(n - 2)\times180^{\circ} $. For a decagon, $ n = 10 $.
## Step2: Calculate the sum of interior angles.
Substitute $ n = 10 $ into the formula: $ S=(10 - 2)\times180^{\circ}=8\times180^{\circ}=1440^{\circ} $.
## Step3: Find the measure of one interior angle.
In a regular polygon, all interior angles are equal. So, one interior angle $ I=\frac{S}{n} $. Substitute $ S = 1440^{\circ} $ and $ n = 10 $: $ I=\frac{1440^{\circ}}{10}=144^{\circ} $.
# Answer:
$ 144^{\circ} $

### 15) Regular 22 - gon (Interior Angle)
# Explanation:
## Step1: Use the sum of interior angles formula.
For a polygon with $ n = 22 $ sides, the sum of interior angles $ S=(n - 2)\times180^{\circ} $.
## Step2: Calculate the sum.
Substitute $ n = 22 $: $ S=(22 - 2)\times180^{\circ}=20\times180^{\circ}=3600^{\circ} $.
## Step3: Find one interior angle.
One interior angle $ I=\frac{S}{n}=\frac{3600^{\circ}}{22}\approx163.6^{\circ} $ (rounded to the nearest tenth).
# Answer:
$ \approx163.6^{\circ} $

### 16) Regular 16 - gon (Interior Angle)
# Explanation:
## Step1: Apply the sum formula.
For $ n = 16 $, sum of interior angles $ S=(16 - 2)\times180^{\circ} $.
## Step2: Compute the sum.
$ S = 14\times180^{\circ}=2520^{\circ} $.
## Step3: Calculate one interior angle.
One interior angle $ I=\frac{2520^{\circ}}{16}=157.5^{\circ} $.
# Answer:
$ 157.5^{\circ} $

### 17) Regular 15 - gon (Interior Angle)
# Explanation:
## Step1: Use the sum formula.
For $ n = 15 $, sum of interior angles $ S=(15 - 2)\times180^{\circ} $.
## Step2: Find the sum.
$ S = 13\times180^{\circ}=2340^{\circ} $.
## Step3: Find one interior angle.
One interior angle $ I=\frac{2340^{\circ}}{15}=156^{\circ} $.
# Answer:
$ 156^{\circ} $

### 18) Regular Heptagon (Exterior Angle)
# Explanation:
## Step1: Recall the property of exterior angles of a polygon.
The sum of exterior angles of any polygon is $ 360^{\circ} $. In a regular polygon, all exterior angles are equal.
## Step2: Find one exterior angle.
For a heptagon, $ n = 7 $. One exterior angle $ E=\frac{360^{\circ}}{n} $. Substitute $ n = 7 $: $ E=\frac{360^{\circ}}{7}\approx51.4^{\circ} $ (rounded to the nearest tenth).
# Answer:
$ \approx51.4^{\circ} $

### 19) Regular 13 - gon (Exterior Angle)
# Explanation:
## Step1: Use the exterior angle sum property.
Sum of exterior angles is $ 360^{\circ} $. For $ n = 13 $, one exterior angle $ E=\frac{360^{\circ}}{n} $.
## Step2: Calculate the exterior angle.
Substitute $ n = 13 $: $ E=\frac{360^{\circ}}{13}\approx27.7^{\circ} $ (rounded to the nearest tenth).
# Answer:
$ \approx27.7^{\circ} $

### 20) Regular Quadrilateral (Exterior Angle)
# Explanation:
## Step1: Recall the exterior angle sum.
Sum of exterior angles of any polygon is $ 360^{\circ} $. A regular quadrilateral (square/rectangle) has $ n = 4 $ sides.
## Step2: Find one exterior angle.
One exterior angle $ E=\frac{360^{\circ}}{n} $. Substitute $ n = 4 $: $ E=\frac{360^{\circ}}{4}=90^{\circ} $.
# Answer:
$ 90^{\circ} $

### 21) Regular Hexagon (Exterior Angle)
# Explanation:
## Step1: Use the exterior angle sum property.
Sum of exterior angles is $ 360^{\circ} $. For a hexagon, $ n = 6 $.
## Step2: Calculate one exterior angle.
One exterior angle $ E=\frac{360^{\circ}}{n} $. Substitute $ n = 6 $: $ E=\frac{360^{\circ}}{6}=60^{\circ} $.
# Answer:
$ 60^{\circ} $

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QUESTION IMAGE

Question

Explanation:

14) Regular Decagon (Interior Angle)

Step1: Recall the formula for the sum of interior angles of a polygon.

Step2: Calculate the sum of interior angles.

Step3: Find the measure of one interior angle.

Step1: Use the sum of interior angles formula.

Step2: Calculate the sum.

Step3: Find one interior angle.

Step1: Apply the sum formula.

Step2: Compute the sum.

Step3: Calculate one interior angle.

Answer:

15) Regular 22 - gon (Interior Angle)