Question

1. a. predict whether each combination of angles in the following table will make a star. if a combination will make a star, predict the number of points it will have.

measure of the interior angle at each point (a)	measure of the exterior angles between adjacent arms (b)	makes a star? (yes/no)	number of points (n)
25°	70°
20°	100°
15°	60°

b. check your predictions by trying to construct each star.

1. give 3 different combinations for angle measures a and b that will form a nine - pointed star.

Explanation:

Step1: Recall the formula

For a star - shaped polygon, the relationship between the interior angle at each point ($a$) and the exterior angle between adjacent arms ($b$) and the number of points ($N$) is given by the formula $b = 2a+\frac{360^{\circ}}{N}$ or $N=\frac{360^{\circ}}{b - 2a}$ (when $b>2a$). If $b\leq2a$, a star - shaped polygon cannot be formed.

Step2: Analyze the first row ($a = 33^{\circ},b = 105^{\circ}$)

Calculate $b-2a$: $b - 2a=105 - 2\times33=105 - 66 = 39^{\circ}$. Then $N=\frac{360}{39}=\frac{120}{13}
otin\mathbb{Z}$ (not an integer). So, it does not make a star.

Step3: Analyze the second row ($a = 25^{\circ},b = 70^{\circ}$)

Calculate $b - 2a$: $b-2a=70 - 2\times25=70 - 50 = 20^{\circ}$. Then $N=\frac{360}{20}=18$. So, it makes a star and has 18 points.

Step4: Analyze the third row ($a = 20^{\circ},b = 100^{\circ}$)

Calculate $b - 2a$: $b-2a=100 - 2\times20=100 - 40 = 60^{\circ}$. Then $N=\frac{360}{60}=6$. So, it makes a star and has 6 points.

Step5: Analyze the fourth row ($a = 15^{\circ},b = 60^{\circ}$)

Calculate $b - 2a$: $b-2a=60 - 2\times15=60 - 30 = 30^{\circ}$. Then $N=\frac{360}{30}=12$. So, it makes a star and has 12 points.

Step6: For forming a nine - pointed star ($N = 9$)

From $N=\frac{360}{b - 2a}$, we have $b-2a = 40^{\circ}$.

• Combination 1: Let $a = 10^{\circ}$, then $b=2a + 40^{\circ}=2\times10+40=60^{\circ}$.
• Combination 2: Let $a = 15^{\circ}$, then $b=2\times15 + 40=70^{\circ}$.
• Combination 3: Let $a = 20^{\circ}$, then $b=2\times20+40 = 80^{\circ}$.

Answer:

Measure of the interior angle at each point ($a$)	Measure of the exterior angles between adjacent arms ($b$)	Makes a star? (yes/no)	Number of points ($N$)
$25^{\circ}$	$70^{\circ}$	yes	18
$20^{\circ}$	$100^{\circ}$	yes	6
$15^{\circ}$	$60^{\circ}$	yes	12

For a nine - pointed star:

• Combination 1: $a = 10^{\circ},b = 60^{\circ}$
• Combination 2: $a = 15^{\circ},b = 70^{\circ}$
• Combination 3: $a = 20^{\circ},b = 80^{\circ}$