coloring activity
directions: solve each problem. circle the answer that matches your solution. color the picture using your selected answers. colors may be used more than once. staple all work to this paper!
1 what is the sum of the measures of the interior angles of a 23 - gon? purple 3,460° orange 3,620° dark green 3,780°
2 what is the measure of each interior angle of a regular 18 - gon? black 140° brown 160° light blue 190°
3 what is the sum of the measures of the exterior angles of a dodecagon? yellow 270° light green 320° red 360°
4 what is the measure of each exterior angle of a regular 20 - gon? pink 18° dark blue 25° orange 162°
5 if the sum of the interior angles of a polygon is 1,980°, how many sides does it have? yellow 13 gray 14 yellow 15
6 if an interior angle of a regular polygon measures 144°, how many sides does it have? light blue 8 purple 9 gray 10
7 if an exterior angle of a regular polygon measures 15°, how many sides does it have? orange 23 dark blue 24 pink 25
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° orange 146° black 154°
9 solve for x

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 $(n - 2)\times180^{\circ}$.
## Step 2: Substitute $n = 23$ into the formula
For a 23 - gon, $n=23$. So we calculate $(23 - 2)\times180^{\circ}$. First, $23-2 = 21$. Then, $21\times180^{\circ}=3780^{\circ}$.
# Answer: Dark Green $3780^{\circ}$

### Problem 2: Measure of each interior angle of a regular 18 - gon
# Explanation:
## Step 1: Find the sum of interior angles
Using the formula $(n - 2)\times180^{\circ}$ with $n = 18$, we get $(18 - 2)\times180^{\circ}=16\times180^{\circ}=2880^{\circ}$.
## Step 2: Divide by the number of angles (sides)
In a regular polygon, all interior angles are equal. So each interior angle $=\frac{(n - 2)\times180^{\circ}}{n}$. For $n = 18$, $\frac{2880^{\circ}}{18}=160^{\circ}$.
# Answer: 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 (including a dodecagon which has 12 sides) is always $360^{\circ}$, regardless of the number of sides.
# Answer: Red $360^{\circ}$

### 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 regular polygon is $360^{\circ}$.
## Step 2: Divide by the number of sides
For a regular 20 - gon, each exterior angle $=\frac{360^{\circ}}{20}=18^{\circ}$.
# Answer: Pink $18^{\circ}$

### Problem 5: Number of sides of a polygon with interior angle sum $1980^{\circ}$
# Explanation:
## Step 1: Use the interior angle sum formula
We know that $(n - 2)\times180^{\circ}=1980^{\circ}$.
## Step 2: Solve for $n$
First, divide both sides by $180^{\circ}$: $n - 2=\frac{1980^{\circ}}{180^{\circ}} = 11$. Then, add 2 to both sides: $n=11 + 2=13$.
# Answer: Yellow $13$

### Problem 6: Number of sides of a regular polygon with interior angle $144^{\circ}$
# Explanation:
## Step 1: Recall the formula for interior angle of a regular polygon
The formula for each interior angle of a regular polygon is $\frac{(n - 2)\times180^{\circ}}{n}$. Let this equal to $144^{\circ}$, so $\frac{(n - 2)\times180^{\circ}}{n}=144^{\circ}$.
## Step 2: Solve for $n$
Multiply both sides by $n$: $(n - 2)\times180^{\circ}=144^{\circ}n$. Expand the left side: $180^{\circ}n-360^{\circ}=144^{\circ}n$. Subtract $144^{\circ}n$ from both sides: $36^{\circ}n-360^{\circ}=0$. Add $360^{\circ}$ to both sides: $36^{\circ}n = 360^{\circ}$. Divide both sides by $36^{\circ}$: $n = 10$. Wait, no, wait: Wait, $(n - 2)\times180=144n$ => $180n-360 = 144n$ => $180n-144n=360$ => $36n = 360$ => $n = 10$? Wait, no, the options are 8, 9, 10. Wait, let's re - check. Wait, if interior angle is $144^{\circ}$, then exterior angle is $180 - 144=36^{\circ}$. And since sum of exterior angles is $360^{\circ}$, number of sides $n=\frac{360}{36}=10$.
# Answer: Gray $10$

### Problem 7: Number of sides of a regular polygon with exterior angle $15^{\circ}$
# Explanation:
## Step 1: Recall the sum of exterior angles
The sum of exterior angles of a regular polygon is $360^{\circ}$.
## Step 2: Solve for $n$
We know that each exterior angle is $15^{\circ}$, so $n=\frac{360^{\circ}}{15^{\circ}} = 24$.
# Answer: Dark Blue $24$

### Problem 8: Remaining angle of a heptagon
# Explanation:
## Step 1: Recall the sum of interior angles of a heptagon
A heptagon has 7 sides. The sum of interior angles is $(7 - 2)\times180^{\circ}=5\times180^{\circ}=900^{\circ}$.
## 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: $127+152 + 112+123+135+105=127+(152 + 112)+(123 + 135)+105=127 + 264+258+105=127+(264 + 258)+105=127 + 522+105=127+105 + 522=232+522 = 754^{\circ}$.
## 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: Orange $146^{\circ}$

Sovi.AI - AI Math Tutor

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

Step 1: Find the sum of interior angles

Step 2: Divide by the number of angles (sides)

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

Answer:

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