Question

Find the value of y and x in the figure

Answer 1

First, find the interior angle missing:

$180-135=45$

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.

$45+(4z+9)+(5y+5)=180\\45+4z+9+5y+5=180\\59+4z+5y=180\\4z+5y=180-59\\4z+5y=121$

After that, we must add the two angles on a straight line on the right-hand side and equate them to 180.

$(4z+9)+(9y-2)=180\\4z+9+9y-2=180\\7+4z+9y=180\\4z+9y=180-7\\4z+9y=173$

We compare the equations:

$4z+5y=121\\4z+9y=173$

We simultaneously solve by isolating one of the variables ($y$) in one of the equations ($4z+5y=121$) and substituting into the other ($4z+9y=173$):

$4z+5y=121\\4z+9y=173\\\\4z+5y=121 => 4z=121-5y\\\\4z+9y=173\\(121-5y)+9y=173\\121-5y+9y=173\\121+4y=173\\4y = 173-121\\4y=52\\y=52/4\\y=13$

We substitute it back into the first formula, the one in which we isolated the variable ($4z+5y=121$):

$y=13\\\\4z+5y=121\\4z+5(13)=121\\4z+65=121\\4z=121-65\\4z=56\\z=56/4\\z=14$

We now check our answer using the formula we know about the internal angles of a triangle ($a+b+c=180$):

$a+b+c=180\\a=45,b=(5y+5),c=(4z+9),y=13,z=14\\45+(5y+5)+(4z+9)=180\\45+5(13)+5+4(14)+9=180\\45+65+5+56+9=180\\180=180$

Now, if $180=180$, then we are correct!

So, in conclusion, $y=13$ and $z = 14$.