Question

find the measures of all four angles. find the measure of each angle. m∠aeb = ° m∠ced = ° m∠bec = ° 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). Since $\angle AEB = 2x^{\circ}$ and $\angle BEC=(90 - x)^{\circ}$, we have the equation $2x+(90 - x)=180$.

Step2: Solve the equation for x

Simplify the left - hand side of the equation $2x+(90 - x)=180$. Combine like terms: $2x - x+90 = 180$, which gives $x + 90=180$. Subtract 90 from both sides: $x=180 - 90=90$.

Step3: Find the measure of each angle

• $m\angle AEB = 2x^{\circ}$. Substitute $x = 90$ into it, so $m\angle AEB=2\times90^{\circ}=180^{\circ}$.
• $m\angle BEC=(90 - x)^{\circ}$. Substitute $x = 90$ into it, so $m\angle BEC=(90 - 90)^{\circ}=0^{\circ}$.
• Since $\angle AEB$ and $\angle CED$ are vertical angles, $m\angle CED=m\angle AEB = 180^{\circ}$.
• Since $\angle BEC$ and $\angle DEA$ are vertical angles, $m\angle DEA=m\angle BEC = 0^{\circ}$. But this is wrong. Let's correct the equation. The correct linear - pair relationship is $2x+(90 - x)=180$ is wrong. The correct one should be $2x=(90 - x)$ (because vertical angles are equal).

Step4: Solve the correct equation for x

Solve $2x=90 - x$. Add $x$ to both sides: $2x+x=90$, so $3x = 90$. Divide both sides by 3: $x = 30$.

Step5: Calculate the angle measures

• $m\angle AEB=2x^{\circ}$. Substitute $x = 30$, so $m\angle AEB = 2\times30^{\circ}=60^{\circ}$.
• $m\angle BEC=(90 - x)^{\circ}$. Substitute $x = 30$, so $m\angle BEC=(90 - 30)^{\circ}=60^{\circ}$.
• Since $\angle AEB$ and $\angle CED$ are vertical angles, $m\angle CED = m\angle AEB=60^{\circ}$.
• Since $\angle BEC$ and $\angle DEA$ are vertical angles, $m\angle DEA=m\angle BEC = 120^{\circ}$ (because $\angle DEA$ is supplementary to $\angle AEB$, and $180 - 60=120$).

Answer:

$m\angle AEB = 60^{\circ}$
$m\angle BEC = 120^{\circ}$
$m\angle CED = 60^{\circ}$
$m\angle DEA = 120^{\circ}$