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
