Question

Thank you very much anh you

Answer 1

Hello from MrBillDoesMath!

Answer:

See discussion below

Discussion:

#4

The sum of the interior angles of a polygon of n sides is (n-2)* 180. As the polygon in #4 has 7 sides , the sum of the interior angles is (7-2)*180 = 5 * 180 = 900.

We can also compute the sum of the interior angles by adding up each of the angles (expressed in terms of x)  shown in #4. Starting at the top of the polygon and proceeding counterclockwise gives:

900 =  (8x +34) + (10x+21)+(9x+30)+(7x+45)+(5x+44)+(12x+13)+(6x+29)  =>

           (8x + 10x+9x+7x+5x+12x+6x) + (34+21+30+45+44+13+29)  =

           57x                                            + 216

This simplifies to

900 =  57x + 216                => subtract 216 from both sides

900-216 = 57x                    => 900-216= 684

684= 57x                            => divide both sides by 57

684/57  = x                         => as 684/57 = 12

x = 12

The diagram has makes repeated use of the same variable for vertices. That is, the diagram shows 4 P's, 1 Q, 2 R's, 0 S's, and 1 T making it impossible to determine the values of some angles.  For example, consider m RST -- whatever that means.

#5 uses the same ideas but the vertices are properly  labeled (I'll leave the grunt of determining the  individual angles to you but here are the main points).

Sum of interior angles = (7 -2) * 180 = 5 * 180 = 900.

900 =  (8x+62) + (7x+66)+(5x+83)+(6x+55)+(10x+42)+(8x+45)+(4x+67)  =>

900 =   (8x + 7x +5x+6x+10x+8x+4x) + (62+66+83+55+42+45+67)

900=     48x                                        +  420             =>

900 -420 = 48x                                                          =>   subtract 420

480  = 48x                                                                  =>   divide by 48

x = 480/48 = 10

Thank you,

MrB