Question

find the value of ( x ), ( y ), and ( z ) in the rhombus below.

(the image shows a rhombus with one angle labeled ( 107^circ ), another angle labeled ( (-x - 8)^circ ), another labeled ( (3y - 1)^circ ), and the last labeled ( (-4z - 7)^circ ))

Explanation:

Step1: Use rhombus adjacent angles property

Adjacent angles in a rhombus are supplementary. So $(-x - 8) + 107 = 180$.
Solve: $-x + 99 = 180 \implies -x = 81 \implies x = -81$.

Step2: Use rhombus opposite angles property

Opposite angles in a rhombus are equal. So $3y - 1 = 107$.
Solve: $3y = 108 \implies y = 36$.

Step3: Use rhombus opposite angles property

Opposite angles in a rhombus are equal. So $-4z - 7 = -x - 8$. Substitute $x=-81$:
$-4z -7 = -(-81) -8 = 73$.
Solve: $-4z = 80 \implies z = -20$.

Answer:

$x=-81$, $y=36$, $z=-20$