Answer
Number of sides: 8, 12, 20, 15, 10, 16, 180
Measure of each exterior $\angle$: 45$^{\circ}$, 30$^{\circ}$, 18$^{\circ}$, 24$^{\circ}$, 36$^{\circ}$, 23.5$^{\circ}$, 2$^{\circ}$
Measure of each interior $\angle$: 135$^{\circ}$, 150$^{\circ}$, 162$^{\circ}$, 156$^{\circ}$, 144$^{\circ}$, 157.5$^{\circ}$, 178$^{\circ}$
Number of diagonals: 20, 54, 170, 90, 35, 104, 15930
Work Step by Step
Use the information already in the table and the following formulas to solve the rest of the table:
Measure of each interior angle (I) where n is the number of sides:
I = ((n-2) $\times$ 180) $\div$ n
Measure of each exterior angle (E) where (S) is the interior angle:
E = 180 - I
Number of diagonals (D) where n is the number of sides:
D = (n(n-3)) $\div$ 2
Example:
We are given 178$^{\circ}$ as the interior measure of one of the angles.
First, we can find the exterior angle.
180$^{\circ}$ - 178$^{\circ}$ = 2$^{\circ}$.
Next, we can find the number of sides.
178$^{\circ}$ = ((n-2) $\times$ 180) $\div$ n, which if we multiply both sides by n is equal to
178$^{\circ}$n = (n-2) $\times$ 180.
178 $\times$ 180 = (180 - 2) $\times$ 180, so there are 180 sides.
Finally, we can find the number of diagonals.
D = (180(180 - 3)) $\div$ 2
D = 15930
