Question

find the value of $x$, $y$, and $z$ in the rhombus below.
$(-y-5)^circ$
$(-10x-8)^circ$
$98^circ$
$(-2z+8)^circ$

Explanation:

Step1: Use rhombus opposite angles property

In a rhombus, opposite angles are equal. So $-10x - 8 = 98$

Step2: Solve for $x$

$$\begin{align*} -10x &= 98 + 8\\ -10x &= 106\\ x &= \frac{106}{-10}\\ x &= -10.6 \end{align*}$$

Step3: Use rhombus consecutive angles property

Consecutive angles in a rhombus are supplementary (sum to $180^\circ$). So $-y - 5 + 98 = 180$

Step4: Solve for $y$

$$\begin{align*} -y + 93 &= 180\\ -y &= 180 - 93\\ -y &= 87\\ y &= -87 \end{align*}$$

Step5: Use rhombus opposite angles property

Opposite angles are equal, so $-2z + 8 = -y - 5$. Substitute $y=-87$: $-2z + 8 = -(-87) - 5$

Step6: Solve for $z$

$$\begin{align*} -2z + 8 &= 87 - 5\\ -2z + 8 &= 82\\ -2z &= 82 - 8\\ -2z &= 74\\ z &= \frac{74}{-2}\\ z &= -37 \end{align*}$$

Answer:

$x = -10.6$, $y = -87$, $z = -37$