Question

Quad. STLM is a rhombus.

If $\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{1}$and $\mathbf{m}\angle \mathbf{3}\mathbf{=}\mathbf{7}\mathbf{x}\mathbf{-}\mathbf{11}$find the value of x.

Short answer

The value of x is 10.

Step-by-step solution

Step 1. Description of step.

Diagonals of a rhombus bisect the vertex angles.

Here, $\angle LSM$is divided into $\angle 1\text{and}\angle \text{2}$, that is, $m\angle 1=m\angle 2$.

Step 2. Description of step.

Here, $\angle SLT$is divided into $\angle 3\text{and}\angle 4$, that is, $m\angle 3=m\angle 4$.

Step 3. Description of step.

Here, $\angle S$and$\angle L$are supplementary.

Substitute $m\angle 3$for $m\angle 4$and $m\angle 1$for $m\angle 2$ to find the value of x.

$\begin{array}{c}m\angle 1+m\angle 2+m\angle 3+m\angle 4=180\\ m\angle 1+m\angle 1+m\angle 3+m\angle 3=180\\ 2m\angle 1+2m\angle 3=180\\ 2\left(m\angle 1+m\angle 3\right)=180\\ 2\left(3x+1+7x-11\right)=180\\ 20\left(x-1\right)=180\\ x-1=9\\ x=10\end{array}$

Therefore, the value of x is 10.