Question

purpose discover a relationship between the number of points on a star and the measures of its interior and exterior angles. common core smp smp 2, smp 3, smp 5, smp 6, smp 7 materials other: a protractor, ruler, and compass grouping work individually getting started the shape of a star is determined by the measure of the interior angle at each point and the measure of an exterior angle formed by adjacent arms. different shape stars can be constructed by using different combinations of angle measures. examples: the measure of the interior angle at each point of this four - pointed star is 40° and the measure of the exterior angle between adjacent arms is 130°. the measure of the interior angle at each point of this five - pointed star is 36° and the measure of the exterior angle is 108°. 1. a. on a separate sheet of paper, construct the stars with the angle measures shown in the table. measure of the interior angle at each point (a) measure of the exterior angles between adjacent arms (b) b - a number of points (n) 30° 150° 20° 92° 20° 80° 15° 60° 8 10° 46° b. complete the last two columns in the table for the stars you constructed in part a.

Explanation:

Step1: Calculate \(b - a\) for each row

For the second row: \(92^{\circ}- 20^{\circ}=72^{\circ}\)
For the third row: \(80^{\circ}-20^{\circ}=60^{\circ}\)
For the fifth row: \(46^{\circ}-10^{\circ}=36^{\circ}\)

Step2: Use the relationship to find the number of points \(N\)

The relationship between the interior - angle \(a\) and the number of points \(N\) of a star is \(a=\frac{180^{\circ}(N - 4)}{N}\) (derived from geometric properties of stars). We can also use another approach. For a regular star, the difference \(b - a\) is related to the central angle subtended by each point. The central angle \(\theta=\frac{360^{\circ}}{N}\). Also, for a star - shaped polygon, we know that there is a geometric relationship. For the first row:
Let's assume the relationship \(b - a=\frac{360^{\circ}}{N}\), when \(b - a = 120^{\circ}\), then \(N=\frac{360^{\circ}}{120^{\circ}} = 3\) (this is a non - standard star case). In general, for a regular star, we can use the fact that the sum of exterior angles of a polygon (star - shaped or not) is \(360^{\circ}\).
For the second row: \(N=\frac{360^{\circ}}{72^{\circ}} = 5\)
For the third row: \(N=\frac{360^{\circ}}{60^{\circ}}=6\)
For the fifth row: \(N=\frac{360^{\circ}}{36^{\circ}} = 10\)

Answer:

Measure of the interior angle at each point \((a)\)	Measure of the exterior angles between adjacent arms \((b)\)	\(b - a\)	Number of points \((N)\)
\(20^{\circ}\)	\(92^{\circ}\)	\(72^{\circ}\)	\(5\)
\(20^{\circ}\)	\(80^{\circ}\)	\(60^{\circ}\)	\(6\)
\(15^{\circ}\)	\(60^{\circ}\)	\(45^{\circ}\)	\(8\)
\(10^{\circ}\)	\(46^{\circ}\)	\(36^{\circ}\)	\(10\)