First, find the interior angle missing:
[tex]180-135=45[/tex]
Then, we know all the interior angles in a triangle add up to 180º. So, we add all the values for the interior angles and we equate them to 180. We solve the equation normally.
[tex]45+(4z+9)+(5y+5)=180\\45+4z+9+5y+5=180\\59+4z+5y=180\\4z+5y=180-59\\4z+5y=121[/tex]
After that, we must add the two angles on a straight line on the right-hand side and equate them to 180.
[tex](4z+9)+(9y-2)=180\\4z+9+9y-2=180\\7+4z+9y=180\\4z+9y=180-7\\4z+9y=173[/tex]
We compare the equations:
[tex]4z+5y=121\\4z+9y=173[/tex]
We simultaneously solve by isolating one of the variables ([tex]y[/tex]) in one of the equations ([tex]4z+5y=121[/tex]) and substituting into the other ([tex]4z+9y=173[/tex]):
[tex]4z+5y=121\\4z+9y=173\\\\4z+5y=121 => 4z=121-5y\\\\4z+9y=173\\(121-5y)+9y=173\\121-5y+9y=173\\121+4y=173\\4y = 173-121\\4y=52\\y=52/4\\y=13[/tex]
We substitute it back into the first formula, the one in which we isolated the variable ([tex]4z+5y=121[/tex]):
[tex]y=13\\\\4z+5y=121\\4z+5(13)=121\\4z+65=121\\4z=121-65\\4z=56\\z=56/4\\z=14[/tex]
We now check our answer using the formula we know about the internal angles of a triangle ([tex]a+b+c=180[/tex]):
[tex]a+b+c=180\\a=45,b=(5y+5),c=(4z+9),y=13,z=14\\45+(5y+5)+(4z+9)=180\\45+5(13)+5+4(14)+9=180\\45+65+5+56+9=180\\180=180[/tex]
Now, if [tex]180=180[/tex], then we are correct!
So, in conclusion, [tex]y=13[/tex] and [tex]z = 14[/tex].
