Question

In the rhombus, angle1 = 18x, angle2 = x + y, and angle3 = 30z. Find the value of the variables x,y, and z. The diagram is not to scale.

Answer 1

You can take z any number that make x and y positive. For example if z =3, x=5 and y=85, then
angle1=90
angle2=90
angle3= 90
hence
angle4=90

Answer 2

This would work too
\[

\begin{array}{ccc}
z=1, & x=\frac{5}{3}, & y=\frac{445}{3}
\end{array}

\]\[
angle1= 30\\
angle2=\frac {450}3 =150
angle3 = 30
angle4 =150

\]
There are infinite solutions for your problem.

Answer 3

\[\begin{array}{ccc}
z=1, & x=\frac{5}{3}, & y=\frac{445}{3}
\end{array}

\]
\[
angle1= 30\\
angle2=\frac {450}3 =150\\
angle3 = 30\\
angle4 =150\\
\]

Answer 4

If you solve these two equations:
\[
18 x+x+y=180\\
30 z=18 x
\]
you get
\[
\begin{array}{cc}
x=\frac{5 z}{3}, & y=180-\frac{95 z}{3}
\end{array}

\]
Take any z that makes y >0 and you get a solution.