Question

find the value of each variable and the measure of each angle
x = 16
y = 45
find the measure of each angle.
m∠bmq=
m∠bmd=
m∠dme=
m∠emq=

Explanation:

Step1: Identify vertical - angle relationships

Vertical angles are equal. $\angle BMQ$ and $\angle DME$ are vertical angles, and $\angle BMD$ and $\angle EMQ$ are vertical angles.
We know that $\angle BMQ=(x + 11)^{\circ}$ and $\angle DME=(4y-x - 11)^{\circ}$, $\angle BMD=(44 - 3y)^{\circ}$ and $\angle EMQ=(4y-x - 11)^{\circ}$. Given $x = 16$ and $y=45$.

Step2: Calculate $\angle BMQ$

Substitute $x = 16$ into the expression for $\angle BMQ$:
$\angle BMQ=(x + 11)^{\circ}=(16 + 11)^{\circ}=27^{\circ}$

Step3: Calculate $\angle DME$

Since $\angle DME=\angle BMQ$ (vertical - angles), $\angle DME = 27^{\circ}$

Step4: Calculate $\angle BMD$

Substitute $y = 45$ into the expression for $\angle BMD$:
$\angle BMD=(44-3y)^{\circ}=(44-3\times45)^{\circ}=(44 - 135)^{\circ}=- 91^{\circ}$, this is wrong. Let's use the fact that $\angle BMD$ and $\angle BMQ$ are supplementary ($\angle BMD+\angle BMQ = 180^{\circ}$)
$\angle BMD=180^{\circ}-\angle BMQ=180 - 27=153^{\circ}$

Step5: Calculate $\angle EMQ$

Since $\angle EMQ=\angle BMD$ (vertical - angles), $\angle EMQ = 153^{\circ}$

Answer:

$m\angle BMQ = 27^{\circ}$
$m\angle BMD = 153^{\circ}$
$m\angle DME = 27^{\circ}$
$m\angle EMQ = 153^{\circ}$