angles of polygons
coloring activity!
name:
date:
directions: solve each problem. circle the answer that matches your solution. color the
your selected answers. colors may be used more than once. staple all work to this pape
1 what is the sum of the measures of the interior angles
of a 23 - gon?
purple
3,460°
orange
3,620°
2 what is the measure of each interior angle of a regular
18 - gon?
black
140°
brown
160°
3 what is the sum of the measures of the exterior
angles of a dodecagon?
yellow
270°
light gre
320°
4 what is the measure of each exterior angle of a regular
20 - gon?
pink
18°
dark bl
25°
5 if the sum of the interior angles of a polygon is 1980°,
how many sides does it have?
yellow
13
gra
14
6 if an interior angle of a regular polygon measures 144°,
how many sides does it have?
light blue
8
pur
7 if an exterior angle of a regular polygon measures 15°,
how many sides does it have?
orange
23
dar
8 if the measures of a heptagon are 127°, 152°, 112°, 123°, 135°,
and 105°, what is the measure of the remaining angle?
light green
138°

Question

Accepted Answer

### Problem 1: Sum of interior angles of a 23 - gon
# Explanation:
## Step 1: Recall the formula for the sum of interior angles of a polygon
The formula for the sum of the interior angles of a polygon with $n$ sides is $S=(n - 2)\times180^{\circ}$. For a 23 - gon, $n = 23$.
## Step 2: Substitute $n = 23$ into the formula
$S=(23 - 2)\times180^{\circ}=21\times180^{\circ}$
$21\times180^{\circ}=3780^{\circ}$? Wait, wait, the options given are $3460^{\circ}$ and $3620^{\circ}$? Wait, maybe I made a mistake. Wait, no, wait the formula is $(n - 2)\times180$. For $n = 23$, $(23-2)\times180=21\times180 = 3780$. But the options are $3460$ and $3620$. Wait, maybe the problem is mis - printed or I misread. Wait, maybe it's a 22 - gon? $(22 - 2)\times180=20\times180 = 3600$, no. 21 - gon: $(21 - 2)\times180=19\times180 = 3420$, no. 23 - gon: $21\times180 = 3780$. Wait, the options given in the picture: Purple $3460^{\circ}$, Orange $3620^{\circ}$. Maybe there is a typo. But according to the formula, the correct sum for a 23 - gon is $(23 - 2)\times180=3780^{\circ}$. But since the options are different, maybe I misread the number of sides. Wait, maybe it's a 20 - gon? No, 20 - gon: $(20 - 2)\times180 = 3240$. Wait, maybe the problem is a 23 - gon, but the options are wrong. Alternatively, maybe the formula is different? No, the formula for the sum of interior angles of a polygon is $(n - 2)\times180$ for $n\geq3$.

### Problem 2: Measure of each interior angle of a regular 18 - gon
# Explanation:
## Step 1: Find the sum of interior angles
Using the formula $S=(n - 2)\times180^{\circ}$, for $n = 18$, $S=(18 - 2)\times180^{\circ}=16\times180^{\circ}=2880^{\circ}$
## Step 2: Find the measure of each interior angle
In a regular polygon, each interior angle $\theta=\frac{(n - 2)\times180^{\circ}}{n}$. For $n = 18$, $\theta=\frac{16\times180^{\circ}}{18}=\frac{2880^{\circ}}{18}=160^{\circ}$. So the answer is Brown $160^{\circ}$

### Problem 3: Sum of exterior angles of a dodecagon
# Explanation:
## Step 1: Recall the property of exterior angles of a polygon
The sum of the exterior angles of any convex polygon (one at each vertex) is always $360^{\circ}$. A dodecagon is a 12 - sided polygon. So the sum of its exterior angles is $360^{\circ}$. But the options are Yellow $270^{\circ}$ and Light Green $320^{\circ}$. This is a contradiction because the sum of exterior angles of any polygon is $360^{\circ}$.

### Problem 4: Measure of each exterior angle of a regular 20 - gon
# Explanation:
## Step 1: Recall the sum of exterior angles
The sum of exterior angles of any convex polygon is $360^{\circ}$.
## Step 2: Find the measure of each exterior angle
In a regular polygon, each exterior angle $\theta=\frac{360^{\circ}}{n}$. For $n = 20$, $\theta=\frac{360^{\circ}}{20}=18^{\circ}$. So the answer is Pink $18^{\circ}$

### Problem 5: Number of sides of a polygon with sum of interior angles $1980^{\circ}$
# Explanation:
## Step 1: Use the sum formula
We know that $S=(n - 2)\times180^{\circ}$, and $S = 1980^{\circ}$. So we set up the equation $(n - 2)\times180^{\circ}=1980^{\circ}$
## Step 2: Solve for $n$
Divide both sides by $180^{\circ}$: $n - 2=\frac{1980^{\circ}}{180^{\circ}} = 11$
Then $n=11 + 2=13$. So the answer is Yellow $13$

### Problem 6: Number of sides of a regular polygon with each interior angle $144^{\circ}$
# Explanation:
## Step 1: Find the measure of each exterior angle
In a regular polygon, the interior angle and exterior angle are supplementary, i.e., $\text{Interior angle}+\text{Exterior angle}=180^{\circ}$. So if the interior angle is $144^{\circ}$, the exterior angle $\theta = 180^{\circ}-144^{\circ}=36^{\circ}$
## Step 2: Find the number of sides
Since the sum of exterior angles is $360^{\circ}$, and each exterior angle is $\theta$, the number of sides $n=\frac{360^{\circ}}{\theta}$. Substituting $\theta = 36^{\circ}$, we get $n=\frac{360^{\circ}}{36^{\circ}} = 10$. But the option is Light Blue $8$. Wait, maybe I made a mistake. Wait, if the interior angle is $144^{\circ}$, then exterior angle is $36^{\circ}$, $n = 360\div36=10$. But the option is $8$. For a regular octagon (8 - gon), each interior angle is $\frac{(8 - 2)\times180}{8}=\frac{1080}{8}=135^{\circ}$. So there is a discrepancy.

### Problem 7: Number of sides of a regular polygon with each exterior angle $15^{\circ}$
# Explanation:
## Step 1: Use the property of exterior angles
The sum of exterior angles of a polygon is $360^{\circ}$. For a regular polygon, the number of sides $n=\frac{360^{\circ}}{\text{measure of each exterior angle}}$
## Step 2: Calculate $n$
Given that each exterior angle is $15^{\circ}$, then $n=\frac{360^{\circ}}{15^{\circ}}=24$. But the option is Orange $23$. So there is a mistake.

### Problem 8: Measure of the remaining angle of a heptagon
# Explanation:
## Step 1: Recall the sum of interior angles of a heptagon
A heptagon has $n = 7$ sides. Using the formula $S=(n - 2)\times180^{\circ}$, $S=(7 - 2)\times180^{\circ}=5\times180^{\circ}=900^{\circ}$
## Step 2: Sum the given angles
The given angles are $127^{\circ},152^{\circ},112^{\circ},123^{\circ},135^{\circ},105^{\circ}$. Let's sum them: $127+152 = 279$; $279+112 = 391$; $391+123 = 514$; $514+135 = 649$; $649+105 = 754$
## Step 3: Find the remaining angle
Let the remaining angle be $x$. Then $754 + x=900$, so $x=900 - 754 = 146^{\circ}$. The option is Light Green $138^{\circ}$, so there is a discrepancy.

Since the user may want solutions to these problems, here are the correct solutions according to the polygon angle formulas:

### Problem 1 (Correct formula - based solution):
# Explanation:
## Step 1: Use the sum of interior angles formula
The formula for the sum of interior angles of a polygon with $n$ sides is $S=(n - 2)\times180^{\circ}$. For a 23 - gon, $n = 23$.
## Step 2: Substitute $n = 23$ into the formula
$S=(23 - 2)\times180^{\circ}=21\times180^{\circ}=3780^{\circ}$
# Answer: $3780^{\circ}$ (Note: The given options may be incorrect)

### Problem 2 (Correct solution):
# Explanation:
## Step 1: Find the sum of interior angles of an 18 - gon
Using $S=(n - 2)\times180^{\circ}$ with $n = 18$, we get $S=(18 - 2)\times180^{\circ}=16\times180^{\circ}=2880^{\circ}$
## Step 2: Find the measure of each interior angle
In a regular polygon, each interior angle $\theta=\frac{S}{n}$. So $\theta=\frac{2880^{\circ}}{18}=160^{\circ}$
# Answer: $160^{\circ}$ (Corresponds to Brown in the options)

### Problem 3 (Correct solution):
# Explanation:
## Step 1: Recall the property of exterior angles
The sum of the exterior angles of any convex polygon (one at each vertex) is $360^{\circ}$. A dodecagon is a convex polygon, so the sum of its exterior angles is $360^{\circ}$
# Answer: $360^{\circ}$ (Note: The given options are incorrect)

### Problem 4 (Correct solution):
# Explanation:
## Step 1: Recall the sum of exterior angles
The sum of exterior angles of any convex polygon is $360^{\circ}$
## Step 2: Find the measure of each exterior angle of a regular 20 - gon
For a regular polygon, each exterior angle $\theta=\frac{360^{\circ}}{n}$. For $n = 20$, $\theta=\frac{360^{\circ}}{20}=18^{\circ}$
# Answer: $18^{\circ}$ (Corresponds to Pink in the options)

### Problem 5 (Correct solution):
# Explanation:
## Step 1: Set up the equation using the sum formula
We know $S=(n - 2)\times180^{\circ}$ and $S = 1980^{\circ}$. So $(n - 2)\times180^{\circ}=1980^{\circ}$
## Step 2: Solve for $n$
Divide both sides by $180^{\circ}$: $n - 2=\frac{1980^{\circ}}{180^{\circ}} = 11$. Then $n=11 + 2=13$
# Answer: $13$ (Corresponds to Yellow in the options)

### Problem 6 (Correct solution):
# Explanation:
## Step 1: Find the exterior angle
In a regular polygon, interior angle + exterior angle $=180^{\circ}$. Given interior angle $=144^{\circ}$, so exterior angle $=180^{\circ}-144^{\circ}=36^{\circ}$
## Step 2: Find the number of sides
The number of sides $n=\frac{360^{\circ}}{\text{exterior angle}}$. So $n=\frac{360^{\circ}}{36^{\circ}} = 10$
# Answer: $10$ (Note: The given option may be incorrect)

### Problem 7 (Correct solution):
# Explanation:
## Step 1: Use the exterior angle - side relationship
The number of sides of a regular polygon $n=\frac{360^{\circ}}{\text{measure of each exterior angle}}$. Given exterior angle $=15^{\circ}$
## Step 2: Calculate $n$
$n=\frac{360^{\circ}}{15^{\circ}}=24$
# Answer: $24$ (Note: The given option may be incorrect)

### Problem 8 (Correct solution):
# Explanation:
## Step 1: Find the sum of interior angles of a heptagon
A heptagon has $n = 7$ sides. Using $S=(n - 2)\times180^{\circ}$, $S=(7 - 2)\times180^{\circ}=5\times180^{\circ}=900^{\circ}$
## Step 2: Sum the given angles
Given angles: $127^{\circ}+152^{\circ}+112^{\circ}+123^{\circ}+135^{\circ}+105^{\circ}=754^{\circ}$
## Step 3: Find the remaining angle
Let the remaining angle be $x$. Then $754^{\circ}+x = 900^{\circ}$, so $x=900^{\circ}-754^{\circ}=146^{\circ}$
# Answer: $146^{\circ}$ (Note: The given option may be incorrect)

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QUESTION IMAGE

Question

Explanation:

Problem 1: Sum of interior angles of a 23 - gon

Step 1: Recall the formula for the sum of interior angles of a polygon

Step 2: Substitute \(n = 23\) into the formula

Problem 2: Measure of each interior angle of a regular 18 - gon

Step 1: Find the sum of interior angles

Step 2: Find the measure of each interior angle

Problem 3: Sum of exterior angles of a dodecagon

Step 1: Recall the property of exterior angles of a polygon

Problem 4: Measure of each exterior angle of a regular 20 - gon

Answer:

Problem 2 (Correct solution):