Each area of a regular polygon measures 45 degrees

Answers

1) part I:         8 sides

 Name:        Octagon

 Part II: 1,080

2) part I:   17

   part II: m<A = 86°

               m<D = 94°

Explanation

Q1) part 1

In a regular polygon,

The sum of exterior angle = 360

let n be the number of sides

∴ 45n = 360

  n = 8 sides

Name: Octagon

part 2

exterior angle + interior angle = 180

Interior angle = 180 - 45°

                      = 135°.

Sum of interior angles = 8 × 135

                                       = 1,080°

Q2

Part I:

The opposite sides of a paralellogram are equal

5x + 2 = 8x - 7

5x - 8x = -7 - 2

  - 3x = - 9

x = 3

AB = CD = 5(3) + 2

                = 15+2

                 = 17

Part II

2y + 50 + 3y + 40 = 180

5y + 90 = 180

5y = 180 - 90

       = 90

y = 90/5

     = 18

Angle A = 2y + 50

              = 2(18) +50

             = 86°

Angle D = 3y + 40

              = 3(18) + 40

                =54 + 40

                  = 94°

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.