Question

Find the value of x and z

Answer 1

we have, the opposite angles at the vertex have the same measure, therefore:

`\begin{gathered} 11x+54=12x+52 \\ 11x+54-54=12x+52-54 \\ 11x=12x-2 \\ 11x-12x=12x-2-12x \\ -x=-2 \\ x=2 \end{gathered}`

then,

`\begin{gathered} angle1\colon11(2)+54=22+54=76 \\ \text{angle}2\colon\text{ 12(}2)+52=24+52=76 \end{gathered}`

and we have the sum of all 4 angles is equal to 360 and as I said before, the opposite angles by the vertex have equal measure, so:

`\begin{gathered} angle\text{ 3 + 76 + 76 + z = 360} \\ \text{angle 3 +152 + z = 360} \\ \text{then, angle 3 = z } \\ z+152+z=360 \\ 2z+152=360 \\ 2z+152-152=360-152 \\ 2z=208 \\ (2z)/(2)=(208)/(2) \\ z=104 \end{gathered}`

answer: x = 2 and z = 104