Question

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. image of a pentagon with angles: (5x + 17)°, 129°, (7x - 46)°, (9x - 11)°, 94° purple 15 light green 16 black 17
10 find m∠c. image of a hexagon - like figure with angles: (8x - 3)° at c, 108° at b, (14x - 25)° at d, 133° at e, (10x - 5)° at a, 52° exterior at f light green 93° red 96° light blue 102°
11 if the figure is a regular polygon, find the value of x. image of a regular polygon with an interior angle (7x + 3)° yellow 19 light blue 20 purple 21
12 if the figure is a regular polygon, find the value of x. image of a regular polygon with an exterior angle (6x - 21)° light blue 11 orange 13 dark blue 15

Explanation:

Step1: Sum of interior angles formula

Sum = $(n-2) \times 180^\circ$, where $n=23$
$\text{Sum}=(23-2)\times180^\circ=21\times180^\circ=3780^\circ$

Step2: Regular interior angle formula

Angle = $\frac{(n-2)\times180^\circ}{n}$, where $n=18$
$\text{Angle}=\frac{(18-2)\times180^\circ}{18}=\frac{16\times180^\circ}{18}=160^\circ$

Step3: Sum of exterior angles rule

Sum of exterior angles of any polygon = $360^\circ$

Step4: Regular exterior angle formula

Angle = $\frac{360^\circ}{n}$, where $n=20$
$\text{Angle}=\frac{360^\circ}{20}=18^\circ$

Step5: Solve for polygon sides

Use $(n-2)\times180^\circ=1980^\circ$
$n-2=\frac{1980^\circ}{180^\circ}=11$, so $n=11+2=13$

Step6: Solve for sides from interior angle

First find exterior angle: $180^\circ-144^\circ=36^\circ$
$n=\frac{360^\circ}{36^\circ}=10$

Step7: Solve for sides from exterior angle

$n=\frac{360^\circ}{15^\circ}=24$

Step8: Find missing heptagon angle

Sum of heptagon interior angles = $(7-2)\times180^\circ=900^\circ$
Sum of given angles: $127+152+112+123+135+105=754^\circ$
Missing angle = $900^\circ-754^\circ=146^\circ$

Step9: Solve for x in pentagon

Sum of pentagon interior angles = $(5-2)\times180^\circ=540^\circ$
Sum of angles: $(5x+17)+129+(7x-46)+(9x-11)+94=540$
Simplify: $21x + 183=540$
$21x=540-183=357$
$x=\frac{357}{21}=17$

Step10: Find $m\angle C$

First, find exterior angle at F: $52^\circ$, so interior $\angle F=180^\circ-52^\circ=128^\circ$
Sum of hexagon interior angles = $(6-2)\times180^\circ=720^\circ$
Set up equation: $(8x-3)+108+(10x-5)+128+133+(14x-25)=720$
Simplify: $32x + 336=720$
$32x=720-336=384$
$x=\frac{384}{32}=12$
$m\angle C=(8\times12-3)^\circ=93^\circ$

Step11: Solve for x in regular 10-gon

First, find interior angle of 10-gon: $\frac{(10-2)\times180^\circ}{10}=144^\circ$
Set $7x+3=144$
$7x=144-3=141$
$x=\frac{141}{7}=19$

Step12: Solve for x in regular octagon

Exterior angle of octagon = $\frac{360^\circ}{8}=45^\circ$
Set $6x-21=45$
$6x=45+21=66$
$x=\frac{66}{6}=11$

Answer:

1. Dark Green $3,780^\circ$
2. Brown $160^\circ$
3. Red $360^\circ$
4. Pink $18^\circ$
5. Yellow 13
6. Gray 10
7. Dark Blue 24
8. Orange $146^\circ$
9. Black 17
10. Light Green $93^\circ$
11. Yellow 19
12. Light Blue 11