Question

How do I solve for x and y using the vertical angle theorem and the triangle sum theorem?

Answer 1

10x = 15y  +  5  (by the vertical angle theorem).
8x + 1 + 7y + 10x = 180   ( by triangle sum theorem)
18x + 7y = 179 .............(1)
from first equation :-
10x - 15y = 5.................(2)

Multiply  (1) by 10 and (2) by -18:-
180x + 70y = 1790
-180x  + 270y = -90
Adding :-
 340y = 1700
y =  5  
and  from first equation
10x = 15(5) + 5 = 80
x = 8 

Answer x=8, y = 5.