Question

find the measures of all four angles.
find the measure of each angle.
m∠aeb =
m∠bec =
m∠ced =
m∠dea =

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle AEB$ and $\angle CED$ are vertical angles, and $\angle BEC$ and $\angle DEA$ are vertical angles. Also, $\angle AEB+\angle BEC = 180^{\circ}$ (linear - pair of angles). So, $4x+(80 - x)=180$.

Step2: Solve the equation for x

Combine like terms: $4x - x+80 = 180$, which simplifies to $3x+80 = 180$. Subtract 80 from both sides: $3x=180 - 80=100$. Then $x=\frac{100}{3}$.

Step3: Find the measure of $\angle AEB$

$\angle AEB = 4x$. Substitute $x = \frac{100}{3}$ into it, so $m\angle AEB=4\times\frac{100}{3}=\frac{400}{3}\approx133.33^{\circ}$.

Step4: Find the measure of $\angle BEC$

$\angle BEC=(80 - x)$. Substitute $x=\frac{100}{3}$ into it, $m\angle BEC=80-\frac{100}{3}=\frac{240 - 100}{3}=\frac{140}{3}\approx46.67^{\circ}$.

Step5: Use vertical - angle equality

Since $\angle AEB=\angle CED$ and $\angle BEC=\angle DEA$ (vertical angles), $m\angle CED=\frac{400}{3}\approx133.33^{\circ}$ and $m\angle DEA=\frac{140}{3}\approx46.67^{\circ}$.

Answer:

$m\angle AEB=\frac{400}{3}^{\circ}$
$m\angle BEC=\frac{140}{3}^{\circ}$
$m\angle CED=\frac{400}{3}^{\circ}$
$m\angle DEA=\frac{140}{3}^{\circ}$