Question

find the value of each variable and the measure of each angle

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, we set up the equation based on the equal - vertical angles. $\angle QME=\angle BMD$ and $\angle QMB=\angle EMD$.
We have the equations:
$4y - x-10=3x - 4$ (from $\angle QME=\angle BMD$) and $x + 10=3x - 4$ (from $\angle QMB=\angle EMD$).
First, solve the equation $x + 10=3x - 4$ for $x$:
Subtract $x$ from both sides: $10=3x - x-4$.

Step2: Simplify the equation for $x$

$10 = 2x-4$. Add 4 to both sides: $10 + 4=2x$, so $14 = 2x$. Divide both sides by 2: $x=\frac{14}{2}=7$.

Step3: Substitute $x = 7$ into the other equation

Substitute $x = 7$ into $4y - x-10=3x - 4$. We get $4y-7 - 10=3\times7 - 4$.
$4y-17 = 21 - 4$.
$4y-17 = 17$.

Step4: Solve for $y$

Add 17 to both sides: $4y=17 + 17$.
$4y=34$. Divide both sides by 4: $y=\frac{34}{4}=8.5$.

Step5: Find the measure of each angle

For $\angle QMB=x + 10$, substitute $x = 7$, then $\angle QMB=7 + 10 = 17^{\circ}$.
$\angle BMD=3x - 4$, substitute $x = 7$, then $\angle BMD=3\times7 - 4=17^{\circ}$.
$\angle QME=4y - x-10$, substitute $x = 7$ and $y = 8.5$, then $\angle QME=4\times8.5-7 - 10=34 - 7 - 10 = 17^{\circ}$.
$\angle EMD=x + 10=17^{\circ}$.

Answer:

$x = 7$
$y = 8.5$